## 1. Inductive Arguments



Let us begin by considering some common kinds of examples of inductive arguments. Consider the following two arguments:




**Example 1.** Every raven in a random sample of 3200
ravens is black. This strongly supports the following conclusion: All
ravens are black.



**Example 2.** 62 percent of voters in a random sample of
400 registered voters (polled on February 20, 2004) said that they
favor John Kerry over George W. Bush for President in the 2004
Presidential election. This supports with a probability of at least
.95 the following conclusion: Between 57 percent and 67 percent of all
registered voters favor Kerry over Bush for President (at or around
the time the poll was taken).




This kind of argument is often called an *induction by
enumeration*. It is closely related to the technique of statistical
estimation. We may represent the logical form of such arguments
semi-formally as follows:




Premise: In random sample *S* consisting of *n* members of
population *B*, the proportion of members that have attribute
*A* is *r*.



Therefore, with degree of support *p*,



Conclusion: The proportion of all members of *B* that have
attribute *A* is between \(r-q\) and \(r+q\) (i.e., lies within
*margin of error* *q* of *r*).




Let's lay out this argument more formally. The premise breaks
down into three separate
 statements:[1]






|  |  |  |
| --- | --- | --- |
|  | **Semi-formalization** | **Formalization** |
| Premise 1 | The frequency (or proportion) of members with attribute *A*
among the members of *S* is *r*. | \(F[A,S] = r\) |
| Premise 2 | *S* is a random sample of *B* with respect to whether
or not its members have *A* |  Rnd[\(S,B,A\)] |
| Premise 3 | Sample *S* has exactly *n* members | Size[\(S] = n\) |
| Therefore | with degree of support *p* | \(========\{p\}\) |
| Conclusion | The proportion of members of *B* that have attribute
*A* lies between \(r-q\) and \(r+q\) (i.e., lies within
*margin of error* *q* of *r*) | \(F[A,B] = r \pm q\) |




Any inductive logic that treats such arguments should address two
challenges. (1) It should tell us which enumerative inductive
arguments should count as *good* inductive arguments. In
particular, it should tell us how to determine the appropriate
*degree* *p* to which such premises *inductively
support* the conclusion, for a given margin of error *q*. (2)
It should demonstrably satisfy the
 CoA.
 That is, it should be provable (as a metatheorem) that *if* a
conclusion expressing the approximate proportion for an attribute in a
population is true, *then* it is very likely that sufficiently
numerous random samples of the population will provide true premises
for *good* inductive arguments that confer *degrees of
support* *p* approaching 1 for that true
conclusion--where, on pain of triviality, these *sufficiently
numerous* samples are only a tiny fraction of a large population.
The supplement on
 Enumerative Inductions: Bayesian Estimation and Convergence,
 shows precisely how a a Bayesian account of enumerative induction may
meet these two challenges.



Enumerative induction is, however, rather limited in scope. This form
of induction is only applicable to the support of claims involving
simple universal conditionals (i.e., claims of form 'All
*B*s are *A*s') and claims about the proportion of an
attribute in a population (i.e., claims of form 'the frequency
of *A*s among the *B*s is *r*'). But, many
important empirical hypotheses are not reducible to this simple form,
and the evidence for these hypotheses is not composed of an
enumeration of such instances. Consider, for example, the Newtonian
Theory of Mechanics:




All objects remain at rest or in uniform motion unless acted upon by
some external force. An object's acceleration (i.e., the rate at
which its motion changes from rest or from uniform motion) is in the
same direction as the force exerted on it; and the rate at which the
object accelerates due to a force is equal to the magnitude of the
force divided by the object's mass. If an object exerts a force
on another object, the second object exerts an equal amount of force
on the first object, but in the opposite direction to the force
exerted by the first object.




The evidence for (and against) this theory is not gotten by examining
a randomly selected subset of objects and the forces acting upon them.
Rather, the theory is tested by calculating what this theory
*says* (or implies) about observable phenomena in a wide
variety of specific situations--e.g., ranging from simple
collisions between small bodies to the trajectories of planets and
comets--and then seeing whether those phenomena occur in the way
that the theory *says* they will. This approach to testing
hypotheses and theories is ubiquitous, and should be captured by an adequate inductive logic.



More generally, for a wide range of cases where inductive
reasoning is important, enumerative induction is inadequate. Rather,
the kind of evidential reasoning that judges the likely truth of hypotheses 
on the basis of what
they *say* (or imply) about the evidence is more appropriate.
Consider the kinds of inferences jury members are supposed to make,
based on the evidence presented at a murder trial. The inference to
probable guilt or innocence is based on a patchwork of evidence of
various kinds. It almost never involves consideration of a randomly
selected sequences of past situations when people like the accused
committed similar murders. Or, consider how a doctor diagnoses her
patient on the basis of his symptoms. Although the frequency of
occurrence of various diseases when similar symptoms have been present may
play a role, this is clearly not the whole story. Diagnosticians
commonly employ a form of *hypothesis evaluation*--e.g.,
would the hypothesis that the patient has a brain tumor account for his symptoms?; or are these symptoms more likely the result of
a minor stroke?; or may some other hypothesis better account for the
patient's symptoms? Thus, a fully adequate account of inductive
logic should explicate the logic of *hypothesis evaluation*,
through which a hypothesis or theory may be tested on the basis of
*what it says* (or "predicts") about observable phenomena. In
 Section 3
 we will see how a kind of probabilistic inductive logic called "Bayesian Inference" or
"Bayesian Confirmation Theory" captures such reasoning. The full logical
structure of such arguments will be spelled out in that section.


## 2. Inductive Logic and Inductive Probabilities



Perhaps the oldest and best understood way of representing partial
belief, uncertain inference, and inductive support is in terms
of *probability* and the equivalent
notion *odds*. Mathematicians have studied probability for over
350 years, but the concept is certainly much older. In recent times a
number of other, related representations of partial belief and
uncertain inference have emerged. Some of these approaches have found
useful application in computer based artificial intelligence systems
that perform inductive inferences in expert domains such as medical
diagnosis. Nevertheless, probabilistic representations have
predominated in such application domains. So, in this article we will
focus exclusively on probabilistic representations of inductive
support. A brief comparative description of some of the most prominent
alternative representations of uncertainty and support-strength can be
found in the supplement
 Some Prominent Approaches to the Representation of Uncertain Inference.


### 2.1 The Historical Origins of Probabilistic Logic



The mathematical study of probability originated with Blaise Pascal
and Pierre de Fermat in the mid-17th century. From that
time through the early 19th century, as the mathematical
theory continued to develop, probability theory was primarily applied
to the assessment of risk in games of chance and to drawing simple
statistical inferences about characteristics of large
populations--e.g., to compute appropriate life insurance premiums
based on mortality rates. In the early 19th century Pierre
de Laplace made further theoretical advances and showed how to apply
probabilistic reasoning to a much wider range of scientific and
practical problems. Since that time probability has become an
indispensable tool in the sciences, business, and many other areas of
modern life.



Throughout the development of probability theory various researchers appear to have thought of it as a kind of logic. But the first extended treatment of
probability as an explicit part of logic was George Boole's
*The Laws of Thought* (1854). John Venn followed two decades
later with an alternative empirical frequentist account of probability
in *The Logic of Chance* (1876). Not long after that the whole
discipline of logic was transformed by new developments in deductive
logic.



In the late 19th and early 20th century Frege,
followed by Russell and Whitehead, showed how deductive logic may be
represented in the kind of rigorous formal system we now call
*quantified predicate logic*. For the
first time logicians had a fully formal deductive logic powerful
enough to represent all valid deductive arguments that arise in
mathematics and the sciences. In this logic the validity of deductive
arguments depends only on the logical structure of the sentences
involved. This development in deductive logic spurred some logicians
to attempt to apply a similar approach to inductive reasoning. The
idea was to extend the deductive entailment relation to a notion of
*probabilistic entailment* for cases where premises provide
less than conclusive support for conclusions. These *partial
entailments* are expressed in terms of *conditional
probabilities*, probabilities of the form \(P[C \pmid B] = r\)
(read "the probability of *C* given *B* is
*r*"), where *P* is a probability function, *C*
is a conclusion sentence, *B* is a conjunction of premise
sentences, and *r* is the probabilistic degree of support that
premises *B* provide for conclusion *C*. Attempts to develop
such a logic vary somewhat with regard to the ways in which they attempt to
emulate the paradigm of formal deductive logic.



Some inductive logicians have tried to follow the deductive paradigm
by attempting to specify inductive support probabilities solely in
terms of the syntactic structures of premise and conclusion sentences.
In deductive logic the syntactic structure of the sentences involved
completely determines whether premises logically entail a conclusion.
So these inductive logicians have attempted to follow suit. 
In such a system each sentence confers a
syntactically specified degree of support on each of the other
sentences of the language. Thus, the inductive probabilities in such a
system are *logical* in the sense that they depend on syntactic
structure alone. This kind of conception was articulated to some
extent by John Maynard Keynes in his *Treatise on Probability*
(1921). Rudolf Carnap pursued this idea with greater rigor in his
*Logical Foundations of Probability* (1950) and in several
subsequent works (e.g., Carnap 1952). (For details of Carnap's
approach see the section on
 logical probability
 in the entry on
 interpretations of the probability calculus,
 in this *Encyclopedia*.) 



In the inductive logics of Keynes and Carnap, Bayes' theorem, a
straightforward theorem of probability theory, plays a central role in
expressing how evidence comes to bear on hypotheses. Bayes'
theorem expresses
how the probability of a hypothesis *h* on the evidence
*e*, \(P[h \pmid e]\), depends on the probability that *e*
should occur if *h* is true, \(P[e \pmid h]\), and on the
probability of hypothesis *h* *prior* to taking the
evidence into account, \(P[h]\) (called the *prior probability*
of *h*). (Later we'll examine Bayes' theorem in detail.) So, such approaches might well be called *Bayesian
logicist* inductive logics. Other prominent Bayesian logicist
attempts to develop a probabilistic inductive logic include the works
of Jeffreys (1939), Jaynes (1968), and Rosenkrantz (1981).



It is now widely held that the core idea of this syntactic approach to
Bayesian logicism is fatally flawed--that syntactic logical
structure cannot be the sole determiner of the degree to which
premises inductively support conclusions. A crucial facet of the
problem faced by syntactic Bayesian logicism involves how the logic is
supposed to apply in scientific contexts where the conclusion sentence
is some scientific hypothesis or theory, and the premises are evidence
claims. The difficulty is that in *any* probabilistic logic
that satisfies the usual axioms for probabilities, the inductive
support for a hypothesis must depend in part on its *prior
probability*. This *prior probability* represents
(arguably) how plausible the hypothesis is taken to be on the basis of
considerations other than the observational and experimental evidence
(e.g., perhaps due to various plausibility arguments). A syntactic
Bayesian logicist must tell us how to assign values to these
pre-evidential *prior probabilities* of hypotheses in a way
that relies only on the syntactic logical structure of the hypothesis,
perhaps based on some measure of syntactic simplicity. There are
severe problems with getting this idea to work. Various
kinds of examples seem to show that such an approach must assign
intuitively quite unreasonable prior probabilities to hypotheses in
specific cases (see the footnote cited near the end of
 Section 3.2
 for details). Furthermore, for this idea to apply to the evidential
support of real scientific theories, scientists would have to
formalize theories in a way that makes their relevant syntactic
structures apparent, and then evaluate theories solely on that
syntactic basis (together with their syntactic relationships to
evidence statements). Are we to evaluate alternative theories of
gravitation, and alternative quantum theories, this way? This seems an
extremely dubious approach to the evaluation of real scientific
hypotheses and theories. Thus, it seems that logical structure alone
may not suffice for the inductive evaluation of scientific hypotheses.
(This issue will be treated in more detail in
 Section 3,
 after we first see how probabilistic logics employ Bayes'
theorem to represent the evidential support for hypotheses as a
function of *prior probabilities* together with
*evidential likelihoods*.)



At about the time that the syntactic Bayesian logicist idea was
developing, an alternative conception of probabilistic inductive
reasoning was also emerging. This approach is now generally referred
to as the Bayesian *subjectivist* or *personalist*
approach to inductive reasoning (see, e.g., Ramsey 1926; De Finetti
1937; Savage 1954; Edwards, Lindman, & Savage 1963; Jeffrey 1983,
1992; Howson & Urbach 1993; Joyce 1999). This approach treats
inductive probability as a measure of an agent's
*degree-of-belief* that a hypothesis is true, given the truth
of the evidence. This approach was originally developed as part of a
larger normative theory of belief and action known as *Bayesian
decision theory*. The principal idea is that the strength of an
agent's desires for various possible outcomes should combine
with her belief-strengths regarding claims about the world to produce
optimally rational decisions. Bayesian subjectivists provide a logic
of decision that captures this idea, and they attempt to justify this
logic by showing that in principle it leads to optimal decisions about
which of various risky alternatives should be pursued. On the Bayesian
subjectivist or personalist account of inductive probability,
inductive probability functions represent the subjective (or personal)
belief-strengths of ideally rational agents, the kind of belief
strengths that figure into rational decision making. (See the section
on
 subjective probability
 in the entry on
 interpretations of the probability calculus,
 in this *Encyclopedia*.)



Elements of a logicist conception of inductive logic live on today as
part of the general approach called *Bayesian inductive logic*.
However, among philosophers and statisticians the term
'Bayesian' is now most closely associated with the
subjectivist or personalist account of belief and decision. And the
term 'Bayesian inductive logic' has come to carry the
connotation of a logic that involves purely subjective probabilities.
This usage is misleading since, for inductive logics, the
Bayesian/non-Bayesian distinction should really turn on whether the
logic gives Bayes' theorem a prominent role, or the approach largely eschews the use of Bayes' theorem in inductive
inferences, as do the *classical approaches* to statistical
inference developed by R. A. Fisher (1922) and by Neyman & Pearson
(1967)). Indeed, any inductive logic that employs the same probability
functions to represent both the *probabilities of evidence claims
due to hypotheses* and the *probabilities of hypotheses due to
those evidence claims* must be a *Bayesian* inductive logic
in this broader sense; because Bayes' theorem follows directly
from the axioms that each probability function must satisfy, and
Bayes' theorem expresses a necessary connection between the
*probabilities of evidence claims due to hypotheses* and the
*probabilities of hypotheses due to those evidence claims*.



In this article the *probabilistic inductive logic* we will
examine is a *Bayesian* inductive logic in this broader sense.
This logic will not presuppose the *subjectivist Bayesian
theory* of belief and decision, and will avoid the objectionable
features of the syntactic version of Bayesian logicism. We will see
that there are good reasons to distinguish *inductive
probabilities* from *degree-of-belief probabilities* and
from *purely syntactic logical probabilities*. So, the
probabilistic logic articulated in this article will be presented in a
way that depends on neither of these conceptions of what the
probability functions *are*. However, this version of the logic
will be general enough that it may be fitted to a Bayesian
subjectivist or Bayesian syntactic-logicist program, if one desires to
do that.


### 2.2 Probabilistic Logic: Axioms and Characteristics



All logics derive from the meanings of terms in sentences. What we now
recognize as *formal deductive logic* rests on the meanings
(i.e., the truth-functional properties) of the standard logical terms.
These logical terms, and the symbols we will employ to represent them,
are as follows: 


* 'not', '\({\nsim}\)';
* 'and', '\(\cdot\)';
* 'inclusive or', '\(\vee\)';
* truth-functional 'if-then', '\(\supset\)';
* 'if and only if', '\(\equiv\)';
* the quantifiers
	+ 'all', '\(\forall\)', and
	+ 'some', '\(\exists\)'; and
* the identity relation, '='.



The meanings of all other terms, the non-logical terms such as names
and predicate and relational expressions, are permitted to
"float free". That is, the logical validity of deductive
arguments depends neither on the meanings of the name and predicate
and relation terms, nor on the truth-values of sentences containing
them. It merely supposes that these non-logical terms are meaningful,
and that sentences containing them have truth-values. Deductive logic
then tells us that the logical structures of some
sentences--i.e., the syntactic arrangements of their logical
terms--preclude them from being jointly true of any possible
state of affairs. This is the notion of *logical
inconsistency*. The notion of *logical entailment* is
inter-definable with it. A collection of premise sentences
*logically entails* a conclusion sentence just when the
negation of the conclusion is *logically inconsistent* with
those premises.



An inductive logic must, it seems, deviate from the paradigm provided
by deductive logic in several significant ways. For one thing, logical
entailment is an absolute, all-or-nothing relationship between
sentences, whereas inductive support comes in degrees-of-strength. For
another, although the notion of *inductive support* is
analogous to the deductive notion of *logical entailment*, and
is arguably an extension of it, there seems to be no inductive logic
extension of the notion of *logical inconsistency*--at
least none that is inter-definable with *inductive support* in
the way that *logical inconsistency* is inter-definable with
*logical entailment*. Indeed, it turns out that when the
unconditional probability of \((B\cdot{\nsim}A)\) is very nearly 0
(i.e., when \((B\cdot{\nsim}A)\) is "nearly
inconsistent"), the degree to which *B* *inductively
supports* *A*, \(P[A \pmid B]\), may range anywhere between 0
and 1. 



Another notable difference is that when *B* *logically
entails* *A*, adding a premise *C* cannot undermine the
logical entailment--i.e., \((C\cdot B)\) must *logically entail*
*A* as well. This property of *logical entailment* is
called *monotonicity*. But *inductive support* is
*nonmonotonic*. In general, depending on what \(A, B\), and
*C* mean, adding a premise *C* to *B* may substantially
raise the degree of support for *A*, or may substantially lower
it, or may leave it completely unchanged--i.e., \(P[A \pmid
(C\cdot B)]\) may have a value much larger than \(P[A \pmid B]\), or
may have a much smaller value, or it may have the same, or nearly the
same value as \(P[A \pmid B]\).



In a formal treatment of probabilistic inductive logic, inductive
support is represented by conditional probability functions defined on
sentences of a formal language *L*. These conditional probability
functions are constrained by certain rules or axioms that are
sensitive to the meanings of the logical terms (i.e.,
'not', 'and', 'or', etc., the
quantifiers 'all' and 'some', and the identity
relation). The axioms apply without regard for what the other terms of
the language may mean. In essence the axioms specify a family of
*possible support functions*, \(\{P\_{\beta}, P\_{\gamma}, \ldots
,P\_{\delta}, \ldots \}\) for a given language *L*. Although each
support function satisfies these same axioms, the further issue of
which among them provides an appropriate measure of *inductive
support* is not settled by the axioms alone. That may depend on
additional factors, such as the meanings of the non-logical terms
(i.e., the names and predicate expressions) of the language.



A good way to specify the axioms of the logic of inductive support
functions is as follows. These axioms are apparently weaker than the
usual axioms for conditional probabilities. For instance, the usual
axioms assume that conditional probability values are restricted to
real numbers between 0 and 1. The following axioms do not assume this,
but only that support functions assign some real numbers as values for
support strengths. However, it turns out that the following axioms
suffice to derive all the usual axioms for conditional probabilities
(including the usual restriction to values between 0 and 1). We draw
on these weaker axioms only to forestall some concerns about whether the support
function axioms may assume too much, or may be overly restrictive.




Let *L* be a language for predicate logic with identity, and let
'\(\vDash\)' be the standard *logical entailment*
relation--i.e., the expression '\(B
\vDash A\)' says
"*B logically entails A*" and the expression '\(\vDash
A\)' says
"*A* is a tautology". A support function is a
function \(P\_{\alpha}\) from pairs of sentences of *L* to real
numbers that satisfies the following axioms:


* (1)\(P\_{\alpha}[E
\pmid F] \ne P\_{\alpha}[G \pmid H]\) for at
least some sentences \(E, F, G\), and *H*.



For all sentence \(A, B, C\), and *D*:


* (2) If
\(B \vDash A\), then \(P\_{\alpha}[A \pmid B] \ge P\_{\alpha}[C \pmid
D]\);
* (3)
\(P\_{\alpha}[A \pmid (B \cdot C)] = P\_{\alpha}[A \pmid (C \cdot B)]\);
* (4) If
\(C \vDash{\nsim}(B \cdot A)\), then either 

\[P\_{\alpha}[(A \vee B) \pmid C] = P\_{\alpha}[A \pmid C] + P\_{\alpha}[B \pmid C]\]

 

 or else \[P\_{\alpha}[E \pmid C] = P\_{\alpha}[C \pmid C]\] for every sentence *E*;
* (5)
\(P\_{\alpha}[(A \cdot B) \pmid C] = P\_{\alpha}[A \pmid (B \cdot C)]
\times P\_{\alpha}[B \pmid C]\).




This axiomatization takes conditional probability as basic, as seems
appropriate for *evidential support functions*. (These
functions agree with the more usual unconditional probability
functions when the latter are defined--just let \(P\_{\alpha}[A] =
P\_{\alpha}[A \pmid (D \vee{\nsim}D)]\). However, these axioms permit
conditional probabilities \(P\_{\alpha}[A \pmid C]\) to remain defined
even when condition statement *C* has probability 0--i.e.,
even when \(P\_{\alpha}[C \pmid (D\vee{\nsim}D)] = 0\).)



Notice that conditional probability functions apply only to pairs of
sentences, a conclusion sentence and a premise sentence. So, in
probabilistic inductive logic we represent finite collections of
premises by conjoining them into a single sentence. Rather than say,





*A* is supported to degree *r* by the set of premises
\(\{B\_1\), \(B\_2\), \(B\_3\),..., \(B\_n\}\), 




we instead say that 




*A* is supported to degree *r* by the conjunctive premise
\((((B\_1\cdot B\_2)\cdot B\_3)\cdot \ldots \cdot B\_n)\),




and write this as 

 

\[P[A \pmid ( ((B\_1\cdot B\_2)\cdot B\_3)\cdot \ldots \cdot B\_n)] = r.\]


The above axioms are quite weak. For instance, they do not say that
logically equivalent sentences are supported by all other sentences to
the same degree; rather, that result is derivable from these axioms
(see
 result 6
 below). Nor do these axioms say that logically equivalent sentences
support all other sentences to the same degree; rather, that result is
also derivable (see
 result 8
 below). Indeed, from these axioms all of the usual theorems of
probability theory may be derived. The following results are
particularly useful in probabilistic logic. Their derivations from
these axioms are provided in note
 2.[2]



1. If \(B \vDash A\), then \(P\_{\alpha}[A \pmid B] =
1\).
2. If \(C \vDash{\nsim}(B\cdot A)\), then either

\[P\_{\alpha}[(A \vee B) \pmid C] = P\_{\alpha}[A \pmid C] + P\_{\alpha}[B \pmid C]\] 

 or else \(P\_{\alpha}[E \pmid C] = 1\) for every sentence
*E*.
3. \(P\_{\alpha}[{\nsim}A \pmid B] = 1 - P\_{\alpha}[A
\pmid B]\) or else \(P\_{\alpha}[C \pmid B] = 1\) for every sentence
*C*.
4. \(1 \ge P\_{\alpha}[A \pmid B] \ge 0\).
5. If \(B \vDash A\), then \(P\_{\alpha}[A \pmid C] \ge
P\_{\alpha}[B \pmid C]\).
6. If \(B \vDash A\) and \(A \vDash B\), then
\(P\_{\alpha}[A \pmid C] = P\_{\alpha}[B \pmid C]\).
7. If \(C \vDash B\), then \(P\_{\alpha}[(A\cdot B)
\pmid C] = P\_{\alpha}[(B\cdot A) \pmid C] = P\_{\alpha}[A \pmid
C]\).
8. If \(C \vDash B\) and \(B \vDash C\), then
\(P\_{\alpha}[A \pmid B] = P\_{\alpha}[A \pmid C]\).
9. \(P\_{\alpha}[B \pmid C] \gt 0\), then 

\[P\_{\alpha}[A \pmid (B\cdot C)] = P\_{\alpha}[B \pmid (A\cdot C)] \times \frac{P\_{\alpha}[A \pmid C]}{P\_{\alpha}[B \pmid C]}\]

(this is a simple form of Bayes' theorem).
10. \(P\_{\alpha}[(A\vee B) \pmid C] = P\_{\alpha}[A
\pmid C] + P\_{\alpha}[B \pmid C] - P\_{\alpha}[(A\cdot B) \pmid C]\).
11. If \(\{B\_1 , \ldots ,B\_n\}\) is any finite set of
sentences such that for each pair \(B\_i\) and \(B\_j, C
\vDash{\nsim}(B\_{i}\cdot B\_{j})\) (i.e., the members of the set are
mutually exclusive, given *C*), then either \(P\_{\alpha}[D
\pmid C] = 1\) for
every sentence *D*, or 

\[
P\_{\alpha}[((B\_1\vee B\_2)\vee \ldots \vee B\_n) \pmid C] = \sum ^{n}\_{i=1} P\_{\alpha}[B\_i \pmid C].
\]
12. If \(\{B\_1 , \ldots ,B\_n , \ldots \}\) is any
countably infinite set of sentences such that for each pair \(B\_i\)
and \(B\_j, C \vDash{\nsim}(B\_{i}\cdot B\_{j})\), then either
\(P\_{\alpha}[D \pmid C] = 1\) for every sentence *D*,
 or[3]
 

\[
\lim\_n P\_{\alpha}[((B\_1\vee B\_2)\vee \ldots \vee B\_n) \pmid C] =
\sum^{\infty}\_{i=1}
P\_{\alpha}[B\_i \pmid C].
\]



Let us now briefly consider each axiom to see how plausible it is as a
constraint on a quantitative measure of inductive support, and how it
extends the notion of deductive entailment. First notice that each
*degree-of-support* function \(P\_{\alpha}\) on *L*
measures *support strength* with some real number values, but
the axioms don't explicitly restrict these values to lie between
0 and 1. It turns out that the all support values must lie between 0
and 1, but this follows from the axioms, rather than being assumed by
them. The scaling of inductive support via the real numbers is surely
a reasonable way to go.



Axiom 1
 is a non-triviality requirement. It says that the support values
cannot be the same for all sentence pairs. This axiom merely rules out
the trivial support function that assigns the same amount of support
to each sentence by every sentence. One might replace this axiom with
the following rule: 

 

\[P\_{\alpha}[(A\vee{\nsim}A) \pmid (A\vee{\nsim}A)] \ne P\_{\alpha}[(A\cdot{\nsim}A) \pmid (A\vee{\nsim}A)].\]


But this alternative rule turns out to be derivable from axiom 1
together with the other axioms.



Axiom 2
 asserts that when *B* *logically entail* *A*, the
support of *A* by *B* is as strong as support can possibly
be. This comports with the idea that an inductive support function is
a generalization of the deductive entailment relation, where the
premises of deductive entailments provide the strongest possible
support for their conclusions.



Axiom 3
 merely says that \((B \cdot C)\) supports sentences to precisely the
same degree that \((C \cdot B)\) supports them. This is an especially
weak axiom. But taken together with the other axioms, it suffices to
entail that logically equivalent sentences support all sentences to
precisely the same degree.



Axiom 4
 says that inductive support *adds up* in a plausible way. When
*C* logically entails the incompatibility of *A* and
*B*, i.e., when no possible state of affairs can make both
*A* and *B* true together, the degrees of support that
*C* provides to each of them individually must sum to the support
it provides to their disjunction. The only exception is in those cases
where *C* acts like a logical contradiction and supports all
sentences to the maximum possible degree (in deductive logic a logical
contradiction *logically entails* every sentence).



To understand what
 axiom 5
 says, think of a support function \(P\_{\alpha}\) as describing a
measure on possible states of affairs. Read each degree-of-support
expression of form '\(P\_{\alpha}[D \pmid E] = r\)' to say
that the proportion of states of affairs in which *D* is true
among those states of affairs where *E* is true is *r*. Read
this way, axiom 5 then says the following. Suppose *B* is true in
proportion *q* of all the states of affairs where *C* is
true, and suppose *A* is true in fraction *r* of those
states where *B* and *C* are true together. Then *A*
and *B* should be true together in what proportion of all the
states where *C* is true? In fraction *r* (the \((A\cdot
B)\) part) of proportion *q* (the *B* portion) of all those
states where *C* is true.



The degree to which a sentence *B* supports a sentence *A*
may well depend on what these sentences mean. In particular it will
usually depend on the meanings we associate with the non-logical terms
(those terms other than the logical terms *not*, *and*,
*or*, etc., the *quantifiers*, and *identity*), that is, on the
meanings of the names, and the predicate and relation terms of the
language. For example, we should want 

 

\[P\_{\alpha}[\textrm{George is not married} \pmid \textrm{George is a bachelor}] = 1,\]


given the usual meanings of 'bachelor' and
'married', since "all bachelors are unmarried"
is analytically true--i.e. no empirical evidence is required to
establish this connection. (In the formal language for predicate
logic, if we associate the meaning "is married" with
predicate term '*M*', the meaning "is a
bachelor" with the predicate term '*B*', and
take the name term '*g*' to refer to George, then we
should want \(P\_{\alpha}[{\nsim}Mg \pmid Bg] = 1\), since \(\forall x
(Bx \supset{\nsim}Mx)\) is analytically true on this meaning
assignment to the non-logical terms.) So, let's associate with
each individual support function \(P\_{\alpha}\) a specific assignment
of meanings (*primary intensions*) to all the non-logical terms
of the language. (However, evidential support functions should not
presuppose meaning assignments in the sense of so-called *secondary
intensions*--e.g., those associated with rigid designators across possible states of affairs. For, we should not want a confirmation function
\(P\_{\alpha}\) to make 

 

\[P\_{\alpha}[\textrm{This glass is full of H\(\_2\)O} \pmid \textrm{This glass is full of water}] = 1,\]


since we presumably want the inductive logic to draw on explicit
empirical evidence to support the claim that water is made of
H2O. Thus, the meanings of terms we associate with a
support function should only be their primary intensions, not their
secondary intensions.)


In the context of inductive logic it
makes good sense to supplement the above axioms with two additional
axioms. Here is the first of them:


* (6) If
*A* is an axiom of set theory or any other piece of pure
mathematics employed by the sciences, or if *A* is
*analytically true* (i.e., if the truth of *A* depends
only on the meanings of the words it contains, where the specific
meanings for names and predicates are those associated with the
particular support function \(P\_{\alpha})\), then, for all sentences
*C*, \(P\_{\alpha}[A \pmid C] = P\_{\alpha}[C \pmid C]\) (i.e.,
\(P\_{\alpha}[A \pmid C] = 1)\).



Here is how axiom 6 applies to the above example, yielding
\(P\_{\alpha}[{\nsim}Mg \pmid Bg] = 1\) when the meaning assignments to
non-logical terms associated with support function \(P\_{\alpha}\)
makes \(\forall x(Bx \supset{\nsim}Mx)\) analytically true. From
axiom 6 (followed by results 7, 5, and 4) we have 

 

\[
\begin{align}
1 & = P\_{\alpha}[\forall x(Bx \supset{\nsim}Mx) \pmid Bg] \\
& = P\_{\alpha}[(Bg \cdot \forall x(Bx \supset{\nsim}Mx)) \pmid Bg]\\
& \le P\_{\alpha}[{\nsim}Mg \pmid Bg] \\
& \le 1;
\end{align}
\]


thus, \(P\_{\alpha}[{\nsim}Mg \pmid Bg] = 1\). The idea behind axiom 6
is that inductive logic is about evidential support for contingent
claims. Nothing can count as empirical evidence for or against
non-contingent truths. In particular, analytic truths should be
maximally supported by all premises *C*. 



One important respect in which inductive logic *should* follow
the deductive paradigm is that the logic should not presuppose the truth of
contingent statements. If a statement *C* is contingent, then some other statements should be able to count as evidence against *C*. Otherwise, a support function \(P\_{\alpha}\) will take *C* and all of its logical consequences to be supported to degree 1 by all possible evidence claims.
This is no way for an inductive logic to behave. The whole idea of inductive logic is
to provide a measure of the extent to which premise statements indicate
the likely truth-values of contingent conclusion statements. This idea
won't work properly if the truth-values of some contingent
statements are *presupposed* by assigning them support value 1 on every possible premise. Such probability assignments would make the inductive logic enthymematic
by hiding significant premises in inductive support relationships.
It would be analogous to permitting deductive arguments to count as valid 
in cases where the explicitly stated premises are insufficient to logically entail the conclusion, but where the validity of the argument is permitted to depend on additional unstated premises. This is not how a
rigorous approach to deductive logic should work, and it should not be a common
practice in a rigorous approach to inductive logic.



Nevertheless, it is common practice for probabilistic logicians to
sweep provisionally accepted contingent claims under the rug by
assigning them probability 1 (regardless of the fact that no explicit
evidence for them is provided). This practice saves
the trouble of repeatedly writing a given contingent sentence *B*
as a premise, since \(P\_{\gamma}[A \pmid B\cdot C]\) will equal
\(P\_{\gamma}[A \pmid C]\) whenever \(P\_{\gamma}[B \pmid C] = 1\).
Although this convention is useful, such probability functions should
be considered mere abbreviations for proper, logically explicit,
non-enthymematic, inductive support relations. Thus, properly
speaking, an inductive support function \(P\_{\alpha}\) should not
assign probability 1 to a sentence on every possible premise unless
that sentence is either (i) logically true, or (ii) an axiom of set
theory or some other piece of pure mathematics employed by the
sciences, or (iii) unless according to the interpretation of the
language that \(P\_{\alpha}\) presupposes, the sentence is
*analytic* (and so outside the realm of evidential support).
Thus, we adopt the following version of the so-called "axiom of
regularity".


* (7) If,
for all *C*, \(P\_{\alpha}[A \pmid C] = P\_{\alpha}[C \pmid C]\)
(i.e., \(P\_{\alpha}[A \pmid C] = 1\)), then *A* must be a logical
truth or an axiom of set theory or some other piece of pure
mathematics employed by the sciences, or *A* must be
*analytically true* (according to the meanings of the terms of
*L* associated with support function \(P\_{\alpha})\).



Axioms 6 and 7 taken together say that a support function
\(P\_{\alpha}\) counts as non-contingently true, and so not subject to
empirical support, just those sentences that are assigned probability
1 by every premise.



Some Bayesian logicists have proposed that an inductive logic might be
made to depend solely on the logical form of sentences, as is the case
for deductive logic. The idea is, effectively, to supplement axioms
1-7 with additional axioms that depend only on the logical
structures of sentences, and to introduce enough such axioms to reduce
the number of possible support functions to a single uniquely best
support function. It is now widely agreed that this project cannot be
carried out in a plausible way. Perhaps support functions should obey
some rules in addition to axioms 1-7. But it is doubtful that
any plausible collection of additional rules can suffice to determine
a single, uniquely qualified support function. Later, in
 Section 3, we will briefly return to this issue,
 after we develop a more detailed account of how inductive probabilities
capture the relationship between hypotheses and evidence.


### 2.3 Two Conceptions of Inductive Probability



Axioms 1-7 for conditional probability functions merely place
formal constraints on what may properly count as a *degree of
support function*. Each function \(P\_{\alpha}\) that satisfies
these axioms may be viewed as a possible way of applying the notion of
*inductive support* to a language *L* that respects the
meanings of the logical terms, much as each possible *truth-value
assignment* for a language represents a possible way of assigning
truth-values to its sentences in a way that respects the meanings of the logical terms. The issue of which
of the *possible* truth-value assignments to a language
represents the *actual* truth or falsehood of its sentences
depends on more than this. It depends on the meanings of the
non-logical terms and on the state of the actual world. Similarly, the
degree to which some sentences *actually* support others in a
fully meaningful language must rely on something more than the mere
satisfaction of the axioms for support functions. It must, at least, rely
on what the sentences of the language mean, and perhaps on much more
besides. But, what more? Perhaps a better understanding of what inductive probability *is* may provide some help by filling out our conception of what
*inductive support* is about. Let's pause to
discuss two prominent views--two *interpretations* of the notion of inductive probability.



One kind of non-syntactic logicist reading of inductive probability takes each support
function \(P\_{\alpha}\) to be a measure on possible states of affairs. The idea is that,
given a fully meaningful language (associated with support function \(P\_{\alpha}\))
'\(P\_{\alpha}[A \pmid B] = r\)' says that among those
states of affairs in which *B* is true, *A* is true in
proportion *r* of them. There will not generally be a single
privileged way to define such a measure on possible states of affairs.
Rather, each of a number of functions \(P\_{\alpha}\), \(P\_{\beta}\),
\(P\_{\gamma}\),..., etc., that satisfy the constraints imposed by
axioms 1-7 may represent a viable measure of the *inferential
import* of the propositions expressed by sentences of the
language. This idea needs more fleshing out, of course. The next
section will provide some indication of how that might
go.



*Subjectivist Bayesians* offer an alternative reading of the
support functions. First, they usually take unconditional probability
as basic, and take conditional probabilities as defined in terms of
unconditional probabilities: the conditional probability
'\(P\_{\alpha}[A \pmid B]\)' is defined as a ratio of
unconditional probabilities: 

 

\[P\_{\alpha}[A \pmid B] = \frac{P\_{\alpha}[A\cdot B]}{P\_{\alpha}[B]}.\]


*Subjectivist Bayesians* take each unconditional probability
function \(P\_{\alpha}\) to represent the belief-strengths or
confidence-strengths of an ideally rational agent, \(\alpha\). On this
understanding '\(P\_{\alpha}[A] =r\)' says, "the
strength of \(\alpha\)'s belief (or confidence) that *A* is
truth is *r*". Subjectivist Bayesians usually tie such
belief strengths to how much money (or how many *units of
utility*) the agent would be willing to bet on *A* turning
out to be true. Roughly, the idea is this. Suppose that an ideally
rational agent \(\alpha\) would be willing to accept a wager that
would yield (no less than) $*u* if *A* turns out to be true
and would lose him $1 if *A* turns out to be false. Then, under
reasonable assumptions about the agent's desire money, it can be
shown that the agent's belief strength that *A* is true
should be 

 

\[P\_{\alpha}[A] = \frac{1}{(u+1)}. \]


And it can further be shown that any function \(P\_{\alpha}\) that
expresses such betting-related belief-strengths on all statements in
agent \(\alpha\)'s language must satisfy axioms for
unconditional probabilities analogous to axioms
 1-5.[4]
 Moreover, it can be shown that any function \(P\_{\beta}\) that
satisfies these axioms is a possible rational belief function for some
ideally rational agent \(\beta\). These relationships between
belief-strengths and the desirability of outcomes (e.g., gaining money
or goods on bets) are at the core of *subjectivist Bayesian
decision theory*. *Subjectivist Bayesians* usually take
*inductive probability* to just *be* this notion of
*probabilistic belief-strength*.



Undoubtedly real agents do believe some claims more strongly than
others. And, arguably, the belief strengths of real agents can be
measured on a probabilistic scale between 0 and 1, at least
approximately. And clearly the inductive support of a hypothesis by
evidence should influence the strength of an agent's belief in
the truth of that hypothesis--that's the point of engaging
in inductive reasoning, isn't it? However, there is good reason
for caution about viewing *inductive support functions* as
Bayesian belief-strength functions, as we'll see a bit later.
So, perhaps an agent's support function is not simply
*identical to* his belief function, and perhaps the
relationship between *inductive support* and
*belief-strength* is somewhat more complicated.



In any case, some account of what support functions are supposed to
represent is clearly needed. The belief function account and the
logicist account (in terms of measures on possible states of affairs)
are two attempts to provide this account. But let us put this interpretative
issue aside for now. One may be able to get a better handle on what
inductive support functions *really are* after one sees how the
inductive logic that draws on them is supposed to work.


## 3. The Application of Inductive Probabilities to the Evaluation of Scientific Hypotheses



One of the most important applications of an inductive logic is its treatment of
the evidential evaluation of scientific hypotheses.
The logic should capture the structure of evidential support for all
sorts of scientific hypotheses, ranging from simple diagnostic claims (e.g.,
"the patient is infected by the HIV") to complex scientific theories about the fundamental nature of the world, such as quantum
mechanics or the theory of relativity. This section will show how
evidential support functions (a.k.a. Bayesian confirmation functions)
represent the evidential evaluation of scientific hypotheses and theories. This logic is essentially comparative. The evaluation of a hypothesis depends on how strongly evidence supports it over alternative hypotheses.



Consider some collection of mutually incompatible, alternative hypotheses (or theories)
about a common subject matter, \(\{h\_1, h\_2 , \ldots \}\). The collection of
alternatives may be very simple, e.g., {"the patient has
HIV", "the patient is free of HIV"}. Or, when the
physician is trying to determine which among a range of diseases is
causing the patient's symptoms, the collection of alternatives may
consist of a long list of possible disease hypotheses. For the cosmologist, the collection of alternatives may consist of several distinct gravitational
theories, or several empirically distinct variants of the "same" theory. Whenever two variants of a hypothesis (or theory) differ in empirical import, they count as distinct hypotheses. (This should not be confused with the converse positivistic assertion that theories with the same empirical content are really the same theory. Inductive logic
doesn't necessarily endorse that view.)



The collection of competing hypotheses (or theories) to be evaluated by the logic may be finite in number, or may be countably infinite. No realistic language contains more than a countable number of expressions; so it suffices for a logic to apply to countably infinite number of sentences. From a purely logical perspective the collection of competing alternatives may consist of every rival hypothesis (or theory) about a given subject matter that can be expressed within a given language -- e.g., all possible theories of the origin and evolution of the universe expressible in English and contemporary mathematics. In practice, alternative hypotheses (or theories) will often be constructed and evidentially evaluated over a long period of time. The logic of evidential support works in much the same way regardless of whether all alternative hypotheses are considered together, or only a few alternative hypotheses are available at a time.



Evidence for scientific hypotheses consists of the results of specific
experiments or observations. For a given experiment or observation,
let '\(c\)' represent a description of the relevant *conditions* under which it is performed, and let
'\(e\)' represent a description of the result of the experiment or observation, the *evidential outcome* of 
conditions \(c\).



The logical connection between scientific hypotheses and the evidence often requires the mediation of background information and auxiliary hypotheses. Let '\(b\)' represent whatever background and auxiliary hypotheses are required to connect each hypothesis \(h\_i\) among the competing hypotheses \(\{h\_1, h\_2 , \ldots \}\) to the evidence. Although the claims expressed by the auxiliary hypotheses within \(b\) may themselves be subject to empirical evaluation, they should be the kinds of claims that
are not at issue in the evaluation of the alternative hypothesis in the collection
\(\{h\_1, h\_2 , \ldots \}\). Rather, each of the alternative hypotheses under consideration draws on the same background and auxiliaries to
logically connect to the evidential events. (If competing hypotheses \(h\_i\) and
\(h\_j\) draw on distinct auxiliary hypotheses \(a\_i\) and \(a\_j\),
respectively, in making logical contact with evidential claims, then
the following treatment should be applied to the respective
conjunctive hypotheses, \((h\_{i}\cdot a\_{i})\) and \((h\_{j}\cdot
a\_{j})\), since these alternative conjunctive hypotheses will
constitute the empirically distinct alternatives at issue.) 



In cases where a hypothesis is deductively related to an
outcome \(e\) of an observational or experimental condition
\(c\) (via background and auxiliaries \(b\)), we will have
either \(h\_i\cdot b\cdot c \vDash
e\) or \(h\_i\cdot b\cdot c
\vDash{\nsim}e\). For example, \(h\_i\) might be the Newtonian
Theory of Gravitation. A test of the theory might involve a condition
statement \(c\) that describes the results of some earlier measurements
of Jupiter's position, and that describes the means by which the
next position measurement will be made; the outcome description
\(e\) states the result of this additional position measurement;
and the background information (and auxiliary hypotheses) \(b\)
might state some already well confirmed theory about the workings and
accuracy of the devices used to make the position measurements. Then,
from \(h\_i\cdot b\cdot c\) we may calculate the specific outcome
\(e\) we expect to find; thus, the following logical entailment
holds: \(h\_i\cdot b\cdot c \vDash
e\). Then, provided that the experimental and observational
conditions stated by \(c\) are in fact true, if the evidential
outcome described by \(e\) actually occurs, the resulting conjoint
evidential claim \((c\cdot e)\) may be considered good evidence for
\(h\_i\), given \(b\). (This method of theory evaluation is called the
*hypothetical-deductive* approach to evidential support.) On
the other hand, when from \(h\_i\cdot b\cdot c\) we calculate some
outcome incompatible with the observed evidential outcome \(e\),
then the following logical entailment holds: \(h\_i\cdot
b\cdot c \vDash{\nsim}e\). In that case, from deductive logic alone we
must also have that \(b\cdot c\cdot e
\vDash{\nsim}h\_i\); thus, \(h\_i\) is said to be
*falsified* by \(b\cdot c\cdot e\). The Bayesian account of
evidential support we will be describing below extends this
deductivist approach to include cases where the hypothesis \(h\_i\)
(and its alternatives) may not be deductive related to the evidence,
but may instead imply that the evidential outcome is likely or unlikely
to some specific degree *r*. That is, the Bayesian approach applies to cases where we may have neither \(h\_i\cdot b\cdot c
\vDash e\) nor \(h\_i\cdot
b\cdot c \vDash{\nsim}e\), but may instead only have \(P[e
\pmid h\_i\cdot b\cdot c] = r\), where *r* is some
"entailment strength" between 0 and 1. 



Before going on to describing the logic of evidential support in more
detail, perhaps a few more words are in order about the background knowledge
and auxiliary hypotheses, represented here by '\(b\)'.
Duhem (1906) and Quine (1953) are generally credited with alerting
inductive logicians to the importance of auxiliary hypotheses in
connecting scientific hypotheses and theories to empirical evidence.
(See the entry on
 Pierre Duhem.)
 They point out that scientific hypotheses often make little contact
with evidence claims on their own. Rather, in most cases scientific hypotheses
make testable predictions only relative to background information and
auxiliary hypotheses that tie them to the evidence. (Some specific examples of such auxiliary hypotheses will be provided in the next subsection.) Typically
auxiliaries are highly confirmed hypotheses from other scientific
domains. They often describe the operating characteristics of various
devices (e.g., measuring instruments) used to make observations or
conduct experiments. Their credibility is usually not at issue in the testing of hypothesis \(h\_i\) against its competitors, because \(h\_i\) and its alternatives
usually rely on the same auxiliary hypotheses to tie them to the
evidence. But even when an auxiliary hypothesis is already
well-confirmed, we cannot simply assume that it is unproblematic, or
just *known to be true*. Rather, the evidential support or
refutation of a hypothesis \(h\_i\) is *relative to* whatever
auxiliaries and background information (in \(b\)) is being
supposed in the confirmational context. In other contexts the auxiliary hypotheses used to test \(h\_i\) may themselves be among a collection of alternative hypotheses
that are subject to evidential support or refutation. Furthermore, to
the extent that competing hypotheses employ different auxiliary
hypotheses in accounting for evidence, the evidence only tests each
such hypothesis in conjunction with its distinct auxiliaries against
alternative hypotheses packaged with their distinct auxiliaries, as
described earlier. Thus, what counts as a *hypothesis to be
tested*, \(h\_i\), and what counts as auxiliary hypotheses and
background information, \(b\), may depend on the epistemic context--on what class of alternative hypotheses are being tested by a collection of experiments or observations, and on what claims are presupposed in that context.
No statement is intrinsically a *test hypothesis*, or
intrinsically an *auxiliary hypothesis* or *background condition*. Rather, these categories are roles statements may play in a particular epistemic context.



In a probabilistic inductive logic the degree to which the evidence
\((c\cdot e)\) supports a hypothesis \(h\_i\) relative to background and auxiliaries
\(b\) is represented by the *posterior probability* of
\(h\_i\), \(P\_{\alpha}[h\_i \pmid b\cdot c\cdot e]\), according to an evidential
support function \(P\_{\alpha}\). It turns out that the *posterior
probability* of a hypothesis depends on just two kinds of factors:
(1) its *prior probability*, \(P\_{\alpha}[h\_i \pmid b]\),
together with the prior probabilities of its competitors,
\(P\_{\alpha}[h\_j \pmid b]\), \(P\_{\alpha}[h\_k \pmid b]\), etc.; and (2) the *likelihood* of evidential outcomes \(e\) according to \(h\_i\) in conjunction with with \(b\) and \(c\), \(P[e \pmid h\_i\cdot b\cdot c]\), together with
the likelihoods of these same evidential outcomes according to competing hypotheses, \(P[e
\pmid h\_j\cdot b\cdot c]\), \(P[e \pmid h\_k\cdot b\cdot c]\), etc. We will now examine each of these factors in some detail. Following that we will see precisely how the values of posterior probabilities depend on the values of likelihoods
and prior probabilities.


### 3.1 Likelihoods



In probabilistic inductive logic *the likelihoods* carry the
empirical import of hypotheses. A *likelihood* is a support
function probability of form \(P[e \pmid h\_i\cdot b\cdot c]\). It
expresses how likely it is that outcome \(e\) will occur according
to hypothesis \(h\_i\) together with the background and auxiliaries \(b\) and the experimental (or observational) conditions \(c\).[5]
 If a hypothesis together with auxiliaries and experimental/observation conditions
deductively entails an evidence claim, the axioms of probability make
the corresponding likelihood *objective* in the sense that every support
function must agree on its values: \(P[e \pmid h\_i\cdot b\cdot c] =
1\) if \(h\_i\cdot b\cdot c \vDash e\); \(P[e \pmid h\_i\cdot b\cdot c]
= 0\) if \(h\_i\cdot b\cdot c \vDash{\nsim}e\). However, in many cases
a hypothesis \(h\_i\) will not be deductively related to the evidence,
but will only imply it probabilistically. There are several ways this
might happen: (1) hypothesis \(h\_i\) may itself be an explicitly
probabilistic or statistical hypothesis; (2) an auxiliary statistical
hypothesis, as part of the background *b*, may connect hypothesis
\(h\_i\) to the evidence; (3) the connection between the hypothesis and
the evidence may be somewhat loose or imprecise, not mediated by
explicit statistical claims, but nevertheless objective enough for the
purposes of evidential evaluation. Let's briefly consider
examples of the first two kinds. We'll treat case (3) in
 Section 5,
 which addresses the the issue of vague and imprecise likelihoods.



The hypotheses being tested may themselves be statistical in nature.
One of the simplest examples of statistical hypotheses and their role
in likelihoods are hypotheses about the chance characteristic of
coin-tossing. Let \(h\_{[r]}\)
be a hypothesis that says a specific coin has a propensity (or
*objective chance*) *r* for coming up *heads* on normal tosses, let \(b\) say that such tosses are probabilistically independent of one another. Let \(c\)
state that the coin is tossed *n* times in the normal way; and
let \(e\) say that on these tosses the coin comes up heads *m*
times. In cases like this the value of the likelihood of the outcome
\(e\) on hypothesis \(h\_{[r]}\)
for condition \(c\) is given by the well-known binomial formula: 

 

\[
P[e \pmid h\_{[r]}\cdot b\cdot c] =
\frac{n!}{m! \times(n-m)!}
\times r^m (1-r)^{n-m}.
\]


There are, of course, more complex cases of likelihoods involving
statistical hypotheses. Consider, for example, the hypothesis that
plutonium 233 nuclei have a half-life of 20 minutes--i.e., that
the propensity (or *objective chance*) for a Pu-233 nucleus to
decay within a 20 minute period is 1/2. The full statistical model for
the lifetime of such a system says that the propensity (or
*objective chance*) for that system to remain intact (i.e., to
*not* decay) within any time period *x* is governed by the
formula \(1/2^{x/\tau}\), where \(\tau\) is the half-life of such a
system. Let \(h\) be a hypothesis that says that this statistical
model applies to Pu-233 nuclei with \(\tau = 20\) minutes; let
\(c\) say that some specific Pu-233 nucleus is intact within a decay detector (of some specific kind) at an initial time \(t\_0\); let \(e\) say that no decay of this same Pu-233 nucleus is detected by the later time \(t\); and let \(b\) say that the detector is completely accurate (it always registers a real decay, and it never registers false-positive detections). Then, the associated likelihood of
\(e\) given \(h\) and \(c\) is this: \(P[e \pmid h\cdot b\cdot c] =
1/2^{(t - t\_0)/\tau}\), where the value of \(\tau\) is 20 minutes.



An auxiliary statistical hypothesis, as part of the background
\(b\), may be required to connect hypothesis \(h\_i\) to the evidence. For example,
a blood test for HIV has a known false-positive rate and a known
true-positive rate. Suppose the false-positive rate is .05--i.e.,
the test tends to incorrectly show the blood sample to be positive for
HIV in 5% of all cases where *HIV is *not* present*. And suppose that the
true-positive rate is .99--i.e., the test tends to correctly show
the blood sample to be positive for HIV in 99% of all cases where *HIV
really is present*. When a particular patient's blood is tested, the hypotheses under consideration are *this patient is infected with HIV*, \(h\), and *this patient is *not* infected with HIV*, \({\nsim}h\). In this context the known test characteristics function as background information, *b*. The
experimental condition \(c\) merely states that this particular
patient was subjected to this specific kind of blood test for HIV,
which was processed by the lab using proper procedures. Let us suppose
that the outcome \(e\) states that the result is a *positive* test
result for HIV. The relevant likelihoods then, are \(P[e \pmid h\cdot
b\cdot c] = .99\) and \(P[e \pmid {\nsim}h\cdot b\cdot c]\) = .05.




In this example the values of the likelihoods are entirely due to the
statistical characteristics of the accuracy of the test, which is
carried by the background/auxiliary information \(b\). The hypothesis
\(h\) being tested by the evidence is not itself statistical.



This kind of situation may, of course, arise for much more complex
hypotheses. The alternative hypotheses of interest may be deterministic
physical theories, say Newtonian Gravitation Theory and some specific alternatives. 
Some of the experiments that test this theory relay on somewhat imprecise
measurements that have known statistical error characteristics, which
are expressed as part of the background or auxiliary hypotheses,
\(b\). For example, the auxiliary \(b\) may describe the error
characteristics of a device that measures the torque imparted to a
quartz fiber, where the measured torque is used to assess the strength
of the gravitational force between test masses. In that case \(b\)
may say that for this kind of device the measurement errors are
normally distributed about whatever value a given gravitational theory
predicts, with some specified standard deviation that is
characteristic of the device. This results in specific values \(r\_i\)
for the likelihoods, \(P[e \pmid h\_i\cdot b\cdot c] = r\_i\), for each
of the various gravitational theories, \(h\_i\), being
tested.



Likelihoods that arise from explicit statistical claims--either
within the hypotheses being tested, or from explicit statistical
background claims that tie the hypotheses to the evidence--are
often called *direct inference likelihoods*. Such likelihoods
should be completely objective. So, all evidential support functions should agree on their values, just as all support functions agree on likelihoods when evidence is logically
entailed. Direct inference likelihoods are *logical* in an
extended, non-deductive sense. Indeed, some logicians have attempted
to spell out the logic of *direct inferences* in terms of the
logical form of the sentences
 involved.[6]
 But regardless of whether that project succeeds, it seems reasonable
to take likelihoods of this sort to have highly objective or
intersubjectively agreed values.



Not all likelihoods of interest in confirmational contexts are
warranted deductively or by explicitly stated statistical claims. In
such cases the likelihoods may have vague, imprecise values, but
values that are determinate enough to still underwrite an objective
evaluation of hypotheses on the evidence. In
 Section 5
 we'll consider such cases, where no underlying statistical
theory is involved, but where likelihoods are determinate enough to
play their standard role in the evidential evaluation of scientific
hypotheses. However, the proper treatment of such cases will be more
easily understood after we have first seen how the logic works when
likelihoods are precisely known (such as cases where the likelihood
values are endorsed by explicit statistical hypotheses and/or explicit
statistical auxiliaries). In any case, the likelihoods that relate
hypotheses to evidence claims in many scientific contexts will have
such objective values. So, although a variety of different support
functions \(P\_{\alpha}\), \(P\_{\beta}\),..., \(P\_{\gamma}\),
etc., may be needed to represent the differing "inductive
proclivities" of the various members of a scientific community,
for now we will consider cases where all evidential support functions
agree on the values of the likelihoods. For,
the likelihoods represent the empirical content of a scientific hypothesis, what
the hypothesis (together with experimental conditions, \(c\), and background and auxiliaries \(b\))
*says* or *probabilistically implies* about the
evidence. Thus, the empirical objectivity of a science relies on a
high degree of objectivity or intersubjective agreement among
scientists on the numerical values of likelihoods.



To see the point more vividly, imagine what a science would be like if
scientists disagreed widely about the values of likelihoods. Each
practitioner interprets a theory to *say* quite different
things about how likely it is that various possible evidence
statements will turn out to be true. Whereas scientist \(\alpha\)
takes theory \(h\_1\) to probabilistically imply that event \(e\) is
highly likely, his colleague \(\beta\) understands the empirical
import of \(h\_1\) to say that \(e\) is very unlikely. And,
conversely, \(\alpha\) takes competing theory \(h\_2\) to
probabilistically imply that \(e\) is very unlikely, whereas
\(\beta\) reads \(h\_2\) to say that \(e\) is extremely likely. So,
for \(\alpha\) the evidential outcome \(e\) supplies strong support
for \(h\_1\) over \(h\_2\), because 

 

\[P\_{\alpha}[e \pmid h\_1\cdot b\cdot c] \gg P\_{\alpha}[e \pmid h\_2\cdot b\cdot c].\]


But his colleague \(\beta\) takes outcome \(e\) to show just the
opposite, that \(h\_2\) is strongly supported over \(h\_1\), because



 \[P\_{\beta}[e \pmid h\_2\cdot b\cdot c] \gg P\_{\beta}[e \pmid h\_1\cdot b\cdot c].\]


If this kind of situation were to occur often, or for significant evidence
claims in a scientific domain, it would make a shambles of the
empirical objectivity of that science. It would completely undermine
the empirical testability of such hypotheses and theories within that
scientific domain. Under these circumstances, although each scientist
employs the same *sentences* to express a given theory
\(h\_i\), each understands the *empirical import* of these
*sentences* so differently that \(h\_i\) as understood by
\(\alpha\) is an empirically different theory than \(h\_i\) as
understood by \(\beta\). (Indeed, arguably, \(\alpha\) must take
at least one of the two sentences, \(h\_1\) or \(h\_2\), to express a different proposition than does \(\beta\).) Thus, the empirical
objectivity of the sciences requires that experts should be in close
agreement about the values of the likelihoods.[7]




For now we will suppose that the likelihoods have objective or
intersubjectively agreed values, common to all agents in a scientific
community. We mark this agreement by dropping the subscript
'\(\alpha\)', '\(\beta\)', etc., from
expressions that represent likelihoods, since all support functions
under consideration are supposed to agree on the values for
likelihoods. One might worry that this supposition is overly strong.
There are legitimate scientific contexts where, although scientists
should have enough of a common understanding of the empirical import
of hypotheses to assign quite similar values to likelihoods, precise
agreement on their numerical values may be unrealistic. This point is
right in some important kinds of cases. So later, in Section 5, we will see how to relax the supposition that precise
likelihood values are available, and see how the logic works in such
cases. But for now the main ideas underlying probabilistic inductive
logic will be more easily explained if we focus on those contexts were
objective or intersubjectively agreed likelihoods are available. Later
we will see that much the same logic continues to apply in contexts
where the values of likelihoods may be somewhat vague, or where
members of the scientific community disagree to some extent about
their values.



An adequate treatment of the likelihoods calls for the introduction of
one additional notational device. Scientific hypotheses are generally
tested by a sequence of experiments or observations conducted over a
period of time. To explicitly represent the accumulation of evidence,
let the series of sentences \(c\_1\), \(c\_2\), ..., \(c\_n\),
describe the conditions under which a sequence of experiments or
observations are conducted. And let the corresponding outcomes of
these observations be represented by sentences \(e\_1\), \(e\_2\),
..., \(e\_n\). We will abbreviate the conjunction of the first
*n* descriptions of experimental or observational conditions by
'\(c^n\)', and abbreviate the conjunction of descriptions
of their outcomes by '\(e^n\)'. Then, for a stream of
*n* observations or experiments and their outcomes, the
likelihoods take form \(P[e^n \pmid h\_{i}\cdot b\cdot c^{n}] = r\),
for appropriate values of \(r\). In many cases the likelihood
of the evidence stream will be equal to the product of the likelihoods
of the individual outcomes:

 

\[
P[e^n \pmid h\_{i}\cdot b\cdot c^{n}] = P[e\_1 \pmid h\_i\cdot b\cdot c\_1] \times \cdots \times P[e\_n \pmid h\_{i}\cdot b\cdot c\_{n}].
\]


When this equality holds, the individual bits of evidence are said to
be *probabilistically independent on the hypothesis (together with
auxiliaries)*. In the following account of the logic of evidential
support, such *probabilistic independence* will *not* be assumed,
except in those places where it is explicitly invoked.


### 3.2 Posterior Probabilities and Prior Probabilities



The probabilistic logic of evidential support represents the net
support of a hypothesis by the *posterior probability of the
hypothesis*, 

\(P\_{\alpha}[h\_i \pmid b\cdot c^{n}\cdot e^{n}]\).

The posterior probability represents the net support for the
hypothesis that results from the evidence, \(c^n \cdot e^n\), together
with whatever *plausibility considerations* are taken to be
relevant to the assessment of \(h\_i\). Whereas the likelihoods are the
means through which evidence contributes to the posterior probability
of a hypothesis, all other relevant plausibility consideration are
represented by a separate factor, called the *prior probability of
the hypothesis*: \(P\_{\alpha}[h\_i \pmid b]\). The *prior
probability* represents the weight of any important considerations
not captured by the evidential likelihoods. Any relevant
considerations that go beyond the evidence itself may be explicitly
stated within expression \(b\) (in addition to whatever auxiliary hypotheses
\(b\) may contain in support of the likelihoods). Thus, the prior probability of \(h\_i\)
may depend explicitly on the content of \(b\). It turns out that posterior
probabilities depend *only* on the values of evidential
likelihoods together with the values of prior probabilities.



As an illustration of the role of *prior probabilities*, consider the
*HIV test* example described in the previous section. What the
physician and the patient want to know is the value of the posterior
probability, \(P\_{\alpha}[h \pmid b\cdot c\cdot e]\), that the patient
has HIV, \(h\), given the evidence of the positive test, \(c\cdot
e\), and given the error rates of the test, described within \(b\).
The value of this posterior probability depends on the likelihood (due
to the error rates) of this patient obtaining a true-positive result,
\(P[e \pmid h\cdot b\cdot c] = .99\), and of obtaining a
false-positive result, \(P[e \pmid {\nsim}h\cdot b\cdot c] = .05\). In
addition, the value of the of the posterior probability depends on how
plausible it is that the patient has HIV prior to taking the test
results into account, \(P\_{\alpha}[h \pmid b]\). In the context of
medical diagnosis, this prior probability is usually assessed on the
basis of the *base rate* for HIV in the patient's risk
group (i.e., whether the patient is an IV drug user, has unprotected sex with
multiple partners, etc.). On a rigorous approach to the logic, such
information and its risk-relevance should be explicitly stated within the
background information \(b\). To see the importance of this
information, consider the following numerical results (which may be
calculated using the formula called Bayes' Theorem, presented in
the next section). If the base rate for the patient's risk group
is relatively high, say \(P\_{\alpha}[h \pmid b] = .10\), then the
positive test result yields a posterior probability value for his
having HIV of \(P\_{\alpha}[h \pmid b\cdot c\cdot e] = .69\). However,
if the patient is in a very low risk group, say \(P\_{\alpha}[h \pmid
b] = .001\), then a positive test result only raises the posterior
probability of his having an HIV infection to \(P\_{\alpha}[h \pmid
b\cdot c\cdot e] = .02\). This posterior probability is much higher
than the prior probability of .001, but should not worry the patient
too much. This positive test result may well be due to the comparatively high
false-positive rate for the test, rather than to the presence of HIV.
This sort of test, with a false-positive rate as large as .05, is
best used as a screening test; a positive result warrants conducting a
second, more rigorous, less error-prone test.



More generally, in the evidential evaluation of scientific hypotheses and theories, prior
probabilities represent assessments of non-evidential *plausibility weightings* among hypotheses. However, because the strengths of such plausibility assessments may
vary among members of a scientific community, critics often brand such assessments as *merely subjective*, and take their role in Bayesian inference to be highly problematic. Bayesian inductivists counter that plausibility
assessments play an important, legitimate role in the sciences, especially
when evidence cannot suffice to distinguish among some alternative hypotheses. And, they argue, the epithet "merely subjective" is unwarranted. Such plausibility assessments are
often backed by extensive arguments that may draw on forceful
conceptual considerations.



Scientists often bring plausibility arguments to bear
in assessing competing views. Although such arguments are seldom
decisive, they may bring the scientific community into widely shared
agreement, especially with regard to the *implausibility* of some
logically possible alternatives. This seems to be the primary
epistemic role of thought experiments.
Consider, for example, the kinds of plausibility arguments that have
been brought to bear on the various interpretations of quantum theory
(e.g., those related to the measurement problem). These arguments go
to the heart of conceptual issues that were central to the original
development of the theory. Many of these issues were first raised by
those scientists who made the greatest contributions to the development of quantum theory, in their attempts to get a conceptual hold on the theory and its implications.



Given any body of evidence, it is fairly easy to cook up
a host of logically possible alternative hypotheses that make the evidence as probable as desired. In particular, it is easy to cook up hypotheses that logically entail any given body evidence, providing likelihood values equal to 1 for all the available evidence. Although most of these cooked up hypotheses will be laughably implausible, evidential likelihoods cannot rule them out. But, the only factors other than likelihoods that figure into the values of posterior probabilities for hypotheses are the values of their prior probabilities; so only prior probability assessments provide a place for the Bayesian logic to bring important plausibility considerations to bear. Thus, the Bayesian logic can only give implausible hypotheses their due via prior probability assessments.



It turns out that the mathematical structure of Bayesian inference makes prior probabilities especially well-suited to represent plausibility assessments among competing hypotheses. For, in the fully fleshed out account of evidential support for hypotheses (spelled out below), it will turn out that only ratios of prior probabilities for competing hypotheses, \(P\_{\alpha}[h\_j \pmid b] / P\_{\alpha}[h\_i \pmid b]\), together with ratios of likelihoods, \(P\_{\alpha}[e \pmid h\_j\cdot b\cdot c] / P\_{\alpha}[e \pmid h\_2\cdot b\cdot c]\), play essential roles. The ratio of prior probabilities is well-suited to represent how much more (or less) plausible hypothesis \(h\_j\) is than competing hypothesis \(h\_i\). Furthermore, the plausibility arguments on which such this comparative assessment is based may be explicitly stated within \(b\). So, given that an inductive logic needs to incorporate well-considered plausibility assessments (e.g. in order to lay low wildly implausible alternative hypotheses), the comparative assessment of Bayesian prior probabilities seems well-suited to do the job.



Thus, although prior probabilities may be subjective in the sense that
agents may disagree on the relative strengths of plausibility
arguments, the priors used in scientific contexts need not
represent *mere subjective whims*. Rather, the comparative strengths of the priors for hypotheses should be supported by arguments about
how much more plausible one hypothesis is than another. The important
role of plausibility assessments is captured by such received bits of
scientific wisdom as the well-known scientific aphorism, *extraordinary claims require
extraordinary evidence*. That is, it takes especially strong
evidence, in the form of extremely high values for (ratios of)
likelihoods, to overcome the extremely low pre-evidential plausibility values
possessed by some hypotheses. In the next section we'll see precisely how this idea works, and we'll return to it again in
 Section 3.4.



When sufficiently strong evidence becomes available, it turns out that the contributions of prior plausibility assessments to the values of posterior probabilities may be substantially "washed
out", overridden by the evidence. That is, provided the prior probability of a true hypothesis isn't assessed to be too
close to zero, the influence of the values of
the prior probabilities will *very probably* fade away as evidence accumulates. In Section 4 we'll see precisely how this kind of Bayesian convergence to the true hypothesis works.
Thus, it turns out that prior plausibility assessments play their most important role
when the distinguishing evidence represented by the likelihoods remains weak.



One more point before moving on to the logic of Bayes' Theorem. Some Bayesian logicists have maintained that posterior
probabilities of hypotheses should be determined by syntactic logical
form alone. The idea is that the likelihoods might reasonably be
specified in terms of syntactic logical form; so if syntactic form
might be made to determine the values of prior probabilities as well,
then inductive logic would be fully "formal" in the same
way that deductive logic is "formal". Keynes and Carnap
tried to implement this idea through syntactic versions of the
principle of indifference--the idea that syntactically similar
hypotheses should be assigned the same prior probability values.
Carnap showed how to carry out this project in detail, but only for
extremely simple formal languages. Most logicians now take the project
to have failed because of a fatal flaw with the whole idea that
reasonable prior probabilities can be made to depend on logical form
alone. Semantic content should matter. Goodmanian grue-predicates
provide one way to illustrate this
 point.[8]
 Furthermore, as suggested earlier, for this idea to apply to the
evidential support of real scientific theories, scientists would have
to assess the prior probabilities of each alternative theory based
only on its syntactic structure. That seems an unreasonable way to
proceed. Are we to evaluate the prior probabilities of alternative
theories of gravitation, or for alternative quantum theories, by
exploring only their syntactic structures, with absolutely no regard
for their content--with *no* regard for what they
*say* about the world? This seems an extremely dubious approach
to the evaluation of real scientific theories. Logical structure alone
cannot, and should not suffice for determining reasonable prior
probability values for real scientific theories. Moreover, real
scientific hypotheses and theories are inevitably subject to
plausibility considerations based on what they *say* about the
world. Prior probabilities are well-suited to represent the comparative weight of plausibility considerations for alternative hypotheses. But no reasonable assessment of comparative plausibility can derive solely from the logical form of hypotheses.



We will return to a discussion of prior probabilities a bit later. Let's now see how Bayesian logic combines likelihoods with prior probabilities
to yield posterior probabilities for hypotheses.


### 3.3 Bayes' Theorem



Any probabilistic inductive logic that draws on the usual
rules of probability theory to represent how evidence supports
hypotheses must be a *Bayesian inductive logic* in the broad
sense. For, Bayes' Theorem follows directly from the usual axioms of probability theory. Its importance derives from the relationship it expresses
between hypotheses and evidence. It
shows how evidence, via the likelihoods, combines with prior
probabilities to produce posterior probabilities for hypotheses.
We now examine several forms of Bayes' Theorem, each derivable from axioms 1-5.


The simplest version of Bayes' Theorem as it applies to evidence for a hypothesis goes like this:




**Bayes' Theorem: Simple Form**

 

\[\begin{align\*}
P\_{\alpha}[h\_i \pmid e] &= \frac{P\_{\alpha}[e \pmid h\_i]\times P\_{\alpha}[h\_i]}{P\_{\alpha}[e]}
\end{align\*}\]


This equation expresses the posterior probability of hypothesis
\(h\_i\) due to evidence \(e\), \(P\_{\alpha}[h\_i \pmid e]\), in terms of the *likelihood* of
the evidence on that hypothesis, \(P\_{\alpha}[e \pmid h\_i]\), the *prior probability of the hypothesis*, \(P\_{\alpha}[h\_i]\), and the *simple probability of the evidence*, \(P\_{\alpha}[e]\). The factor \(P\_{\alpha}[e]\) is often called *the expectedness of the evidence*. Written this way, the theorem suppresses the experimental (or observational) conditions, \(c\), and all background information and auxiliary hypotheses, \(b\). As discussed earlier, both of these terms play an important role in logically connecting the hypothesis at issue, \(h\_i\), to the evidence \(e\). In scientific contexts the objectivity of the likelihoods, \(P\_{\alpha}[e \pmid h\_i\cdot b \cdot c]\), almost always depends on such terms. So, although the suppression of experimental (or observational) conditions and auxiliary hypotheses is a common practice in accounts of Bayesian inference, the treatment below, and throughout the remainder of this article will make the role of these terms explicit.



The subscript \(\alpha\) on the evidential support function \(P\_{\alpha}\) is there to remind us that more than one such function exists. A host of distinct probability functions satisfy axioms 1-5, so each of them satisfies Bayes' Theorem. Some of these probability functions may provide a better fit with our intuitive conception of how the evidential support for hypotheses should work. Nevertheless, there are bound to be reasonable differences among Bayesian agents regarding to the initial plausibility of a hypothesis \(h\_i\). This diversity in initial plausibility assessments is represented by diverse values for prior probabilities for the hypothesis: \(P\_{\alpha}[h\_i]\), \(P\_{\beta}[h\_i]\), \(P\_{\gamma}[h\_i]\), etc. This usually results in diverse values for posterior probabilities for hypotheses: \(P\_{\alpha}[h\_i \pmid e]\), \(P\_{\beta}[h\_i \pmid e]\), \(P\_{\gamma}[h\_i \pmid e]\), etc. So it is important to keep the diversity among evidential support functions in mind.



Here is how the Simple Form of Bayes' Theorem looks
when terms for the experimental (or observational) conditions, \(c\), and the
background information and auxiliary hypotheses \(b\) are made explicit:




**Bayes' Theorem: Simple Form with explicit Experimental Conditions, Background Information and Auxiliary Hypotheses**

 

\[\tag{8}
\begin{align}
 P\_{\alpha}[h\_i \pmid b\cdot c\cdot e]
&=
\frac{P[e \pmid h\_i\cdot b \cdot c] \times P\_{\alpha}[h\_i \pmid b]}
{P\_{\alpha}[e \pmid b \cdot c]}

\\ &\qquad
\times

\frac{P\_{\alpha}[c \pmid h\_i\cdot b]}
{P\_{\alpha}[c \pmid b]}\\[3ex]
& =
\frac{P[e \pmid h\_i\cdot b\cdot c] \times P\_{\alpha}[h\_i \pmid b]}
{P\_{\alpha}[e \pmid b\cdot c]}\\[2ex]
& \textrm{when }
P\_{\alpha}[c \pmid h\_j\cdot b] =
P\_{\alpha}[c \pmid b].

\end{align}
\]


This version of the theorem determines the posterior probability of the hypothesis,
\(P\_{\alpha}[h\_i \pmid b\cdot c\cdot e]\), from the value of the
*likelihood* of the evidence according to that hypothesis (taken together with
background and auxiliaries and the experimental conditions), \(P[e \pmid h\_i\cdot b\cdot c]\), the value of the *prior probability of the hypothesis* (on background and auxiliaries), \(P\_{\alpha}[h\_i \pmid b]\), and the value of the *expectedness of the evidence* (on background and auxiliaries and the experimental conditions), \(P\_{\alpha}[e \pmid b\cdot c]\). Notice that in the factor for the likelihood, \(P[e \pmid h\_i\cdot b\cdot c]\), the subscript \(\alpha\) has been dropped. This marks the fact that in scientific contexts the likelihood of an evidential outcome \(e\) on the hypothesis together with explicit background and auxiliary hypotheses and the description of the experimental conditions, \(h\_i\cdot b\cdot c\), is usually objectively determinate. This factor represents what the hypothesis (in conjunction with background and auxiliaries) *objectively says* about the likelihood of possible evidential outcomes of the experimental conditions. So, all reasonable support functions should agree on the values for likelihoods. (Section 5 will treat cases where the likelihoods may lack this kind of objectivity.)



This version of Bayes' Theorem includes a term that represents the ratio of the *likelihood of the experimental conditions* on the hypothesis and background information (and auxiliaries) to the
*"likelihood" of the experimental conditions* on
the background (and auxiliaries) alone:

\(P\_{\alpha}[c \pmid h\_i\cdot b]/ P\_{\alpha}[c \pmid b]\).

Arguably the value of this term should be 1, or very nearly 1, since the
truth of the hypothesis at issue should not significantly affect how
likely it is that the experimental conditions are satisfied. If
various alternative hypotheses assign significantly different
likelihoods to the experimental conditions themselves, then such
conditions should more properly be included as part of the evidential
outcome \(e\).



Both the *prior probability* of the hypothesis and the
*expectedness* tend to be somewhat subjective factors in that
various agents from the same scientific community may legitimately
disagree on what values these factors should take. Bayesian logicians
usually accept the apparent subjectivity of the prior probabilities of
hypotheses, but find the subjectivity of the *expectedness* to
be more troubling. This is due at least in part to the fact that in a
Bayesian logic of evidential support the value of the expectedness
cannot be determined independently of likelihoods and prior
probabilities of hypotheses. That is, when, for each member of a collection
of alternative hypotheses, the likelihood \(P[e \pmid h\_j\cdot b\cdot
c]\) has an objective (or intersubjectively agreed) value, the
*expectedness* is constrained by the following equation (where
the sum ranges over a mutually exclusive and exhaustive collection of
alternative hypotheses \(\{h\_1, h\_2 , \ldots ,h\_m , \ldots \}\), which
may be finite or countably infinite):




\[
\begin{align}
P\_{\alpha}[e \pmid b\cdot c] &= \sum\_j P[e \pmid h\_j\cdot b\cdot c] \times P\_{\alpha}[h\_j \pmid b \cdot c].
\end{align}
\]


This equation shows that the values for the prior probabilities
together with the values of the likelihoods uniquely determine the
value for the *expectedness of the evidence*. Furthermore, it
implies that the value of *the expectedness* must lie between
the largest and smallest of the various likelihood values implied by
the alternative hypotheses. However, the precise value of *the
expectedness* can only be calculated this way when every
alternative to hypothesis \(h\_j\) is specified. In cases where some
alternative hypotheses remain unspecified (or undiscovered), the value
of *the expectedness* is constrained in principle by the
totality of possible alternative hypotheses, but there is no way to
figure out precisely what its value should be.



Troubles with determining a numerical value for the *expectedness of the evidence*
may be circumvented by appealing to another form of Bayes'
Theorem, a ratio form that compares hypotheses one pair at a time:




**Bayes' Theorem: Ratio Form**

 

\[\tag{9}
\begin{align}
\frac{P\_{\alpha}[h\_j \pmid b\cdot c\cdot e]}
{P\_{\alpha}[h\_i \pmid b\cdot c\cdot e]}
& =
\frac{P[e \pmid h\_j\cdot b\cdot c]}
{P[e \pmid h\_i\cdot b\cdot c]}
\times
\frac{P\_{\alpha}[h\_j \pmid b]}
{P\_{\alpha}[h\_i \pmid b]}\\
&\qquad\times
\frac{P\_{\alpha}[c \pmid h\_j\cdot b]}
{P\_{\alpha}[c \pmid h\_i\cdot b]}\\[2ex]
& =
\frac{P[e \pmid h\_j\cdot b\cdot c]}
{P[e \pmid h\_i\cdot b\cdot c]}
 \times
\frac{P\_{\alpha}[h\_j \pmid b]}
{P\_{\alpha}[h\_i \pmid b]}
\\[2ex]
& \textrm{when }
P\_{\alpha}[c \pmid h\_j\cdot b] =
P\_{\alpha}[c \pmid h\_i\cdot b].

\end{align}
\]



The clause 

\(P\_{\alpha}[c \pmid h\_j\cdot b] = P\_{\alpha}[c \pmid h\_i\cdot b]\)

says that the experimental (or observation) condition described by \(c\) is as likely on \((h\_i\cdot b)\) as on \((h\_j\cdot b)\) -- i.e., the experimental or observation conditions are no more likely according to one hypothesis than according to the other.[9]




This Ratio Form of Bayes' Theorem expresses how much more
plausible, on the evidence, one hypothesis is than another. Notice
that the *likelihood ratios* carry the full import of the
evidence. The evidence influences the evaluation of hypotheses in no
other way. The only other factor that influences the value of the
ratio of posterior probabilities is the ratio of the prior
probabilities. When the likelihoods are fully objective, any
subjectivity that affects the ratio of posteriors can only arise via
subjectivity in the ratio of the priors. 



This version of Bayes's Theorem shows that in order to evaluate
the *posterior probability ratios* for pairs of hypotheses, the
prior probabilities of hypotheses need not be evaluated absolutely;
only their ratios are needed. That is, with regard to the priors, the
Bayesian evaluation of hypotheses only relies on *how much more
plausible* one hypothesis is than another (due to considerations
expressed within *b*). This kind of Bayesian evaluation of
hypotheses is essentially comparative in that only *ratios of
likelihoods* and *ratios of prior probabilities* are ever
really needed for the assessment of scientific hypotheses.
Furthermore, we will soon see that the absolute values of the
posterior probabilities of hypotheses entirely derive from the
*posterior probability ratios* provided by the Ratio Form of
Bayes' Theorem. 



When the evidence consists of a collection of *n* distinct
experiments or observations, we may explicitly represent this fact by
replacing the term '\(c\)' by the conjunction of experimental or observational conditions, \((c\_1\cdot
c\_2\cdot \ldots \cdot c\_n)\), and replacing the term
'\(e\)' by the conjunction of their respective outcomes, \((e\_1\cdot e\_2\cdot \ldots \cdot e\_n)\). For notational convenience, let's use the term
'\(c^n\)' to abbreviate the conjunction of *n* the experimental conditions, and we use the term '\(e^n\)' to abbreviate the corresponding conjunction of *n* their respective outcomes. Relative to any given hypothesis \(h\), the evidential
outcomes of distinct experiments or observations will usually be
probabilistically independent of one another, and also independent of the
experimental conditions for one another. In that case we have:

 

\[
P[e^n \pmid h\cdot b\cdot c^n] = P[e\_1 \pmid h\cdot b\cdot c\_1] \times \cdots \times P[e\_n \pmid h\cdot b\cdot c\_n].
\]


When the Ratio Form of Bayes' Theorem is extended to explicitly represent the evidence as consisting of a collection of *n* of distinct experiments (or observations) and their respective outcomes, it takes the following form. 




**Bayes' Theorem: Ratio Form for a Collection of *n*
Distinct Evidence Claims**

 
\[\tag{9\*}
\begin{align}

\frac{P\_{\alpha}[h\_j \pmid b\cdot c^n \cdot e^n ] }
{P\_{\alpha}[h\_i \pmid b\cdot c^n \cdot e^n ]}
& =
\frac{P [ e^n \pmid h\_j\cdot b\cdot c^n ]}
{P [ e^n \pmid h\_i\cdot b\cdot c^n ]}
\times
\frac{P\_{\alpha}[h\_j \pmid b]}
{P\_{\alpha}[h\_i \pmid b]}\\[2ex]
&\qquad \times
\frac{P\_{\alpha} [ c^n \pmid h\_j\cdot b]}
{P\_{\alpha} [ c^n \pmid h\_i\cdot b]}\\[2ex]
& =
\frac{P [ e^n \pmid h\_j\cdot b\cdot c^n ]}
{P [ e^n \pmid h\_i\cdot b\cdot c^n ]}
\times
\frac{P\_{\alpha}[h\_j \pmid b]}
{P\_{\alpha}[h\_i \pmid b]}
\\[2ex]
&\textrm{when }
P\_{\alpha} [ c ^n \pmid h \_j\cdot b ] =
P\_{\alpha} [ c ^n \pmid h \_i\cdot b ].
\end{align}
\]

Furthermore, when evidence claims are probabilistically independent of one another, we have




\[\tag{9\*\*}
\begin{align}

\frac{P\_{\alpha}[h\_j \pmid b\cdot c^n \cdot e^n ] }
{P\_{\alpha}[h\_i \pmid b\cdot c^n \cdot e^n ]}
& =
\frac{P[e\_1 \pmid h\_j\cdot b\cdot c\_1]}
{P[e\_1 \pmid h\_i\cdot b\cdot c\_1]}
\times \cdots \\[2ex]
&\qquad \times
\frac{P[e\_n \pmid h\_{j }\cdot b\cdot c\_{ n}]}
{P[e\_n \pmid h\_{i }\cdot b\cdot c\_{ n}]}
\times
\frac{P\_{\alpha}[h\_j \pmid b]}
{P\_{\alpha}[h\_i \pmid b]}.

\end{align}
\]


Let's consider a simple example of how the Ratio Form of
Bayes' Theorem applies to a collection of independent evidential events. Suppose we possess a warped coin
and want to determine its propensity for *heads* when tossed in
the usual way. Consider two hypotheses, \(h\_{[p]}\) and
\(h\_{[q]}\), which say that the propensities for the coin to come up
*heads* on the usual kinds of tosses are \(p\) and \(q\),
respectively. Let \(c^n\) report that the coin is tossed *n*
times in the normal way, and let \(e^n\) report that precisely
*m* occurrences of *heads* has resulted. Supposing that
the outcomes of such tosses are probabilistically independent (asserted by \(b\)),
the respective likelihoods take the binomial form 

 

\[
P[e^n \pmid h\_{[r]}\cdot b\cdot c^n] =
\frac{n!}{m! \times(n-m)!}
\times r^m (1-r)^{n-m},
\]
 
with \(r\) standing in for \(p\) and for \(q\), respectively. Then, Equation 9\*\*
 yields the following formula, where the likelihood ratio is the
ratio of the respective binomial terms:

 

\[
\frac{P\_{\alpha}[h\_{[p]} \pmid b\cdot c^{n }\cdot e^{ n}]}
{P\_{\alpha}[h\_{[q]} \pmid b\cdot c^{n }\cdot e^{ n}]}
=
\frac{p^m (1-p)^{n-m}}
{q^m (1-q)^{n-m}}
\times
\frac{P\_{\alpha}[h\_{[p]} \pmid b]}
{P\_{\alpha}[h\_{[q]} \pmid b]}
\]


When, for instance, the coin is tossed \(n = 100\) times and comes up
*heads* \(m = 72\) times, the evidence for hypothesis
\(h\_{[1/2]}\) as compared to \(h\_{[3/4]}\) is given by the likelihood
ratio

 

\[\frac{P [ e^n \pmid h\_{[1/2]}\cdot b\cdot c^n ]}
{P [ e^n \pmid h\_{[3/4]}\cdot b\cdot c^n ]} = \frac{[(1/2)^{72}(1/2)^{28}]}{[(3/4)^{72}(1/4)^{28}]} = .000056269. \]


In that case, even if the prior plausibility considerations
(expressed within \(b\)) make it 100 times more plausible that the
coin is *fair* than that it is warped towards *heads* with
*propensity 3/4* -- i.e., even if \(P\_{\alpha}[h\_{[1/2]} \pmid b] / P\_{\alpha}[h\_{[3/4]} \pmid b] = 100\) -- the evidence provided by these tosses makes the posterior plausibility that the coin is *fair*
only about 6/1000ths as plausible as the hypothesis that it
is warped towards *heads* with *propensity 3/4*:



\[
\frac{P\_{\alpha}[h\_{[1/2]} \pmid b\cdot c^{n}\cdot e^{n}]}{P\_{\alpha}[h\_{[3/4]} \pmid b\cdot c^{n}\cdot e^{n}]} = .0056269.
\]


Thus, such evidence *strongly refutes* the "fairness
hypothesis" relative to the "3/4-*heads*
hypothesis", provided the assessment of prior
prior plausibilities doesn't make the latter hypothesis *too
extremely implausible* to begin with. Notice, however, that
*strong refutation* is not *absolute refutation*.
Additional evidence could reverse this trend towards the
refutation of the *fairness hypothesis*.



This example employs repetitions of the same kind of
experiment--repeated tosses of a coin. But the point holds more
generally. If, as the evidence increases, the *likelihood
ratios* 

 

\[\frac{P[e^n \pmid h\_{j}\cdot b\cdot c^{n}]}{P[e^n \pmid h\_{i}\cdot b\cdot c^{n}]}\]


approach 0, then the Ratio Forms of Bayes' Theorem, Equations \(9\*)\) and \(9\*\*)\), 
 show that the posterior probability of \(h\_j\) must approach 0 as
well, since 

 

\[P\_{\alpha}[h\_j \pmid b\cdot c^{n}\cdot e^{n}] \le \frac{P\_{\alpha}[h\_j \pmid b\cdot c^{n}\cdot e^{n}]}{P\_{\alpha}[h\_i \pmid b\cdot c^{n}\cdot e^{n}]}.\]


Such evidence comes to strongly refute \(h\_j\), with little regard for
its prior plausibility value. Indeed, Bayesian induction turns out to
be a version of *eliminative induction*, and Equation \(9\*\) and \(9\*\*\) begin
to illustrate this. For, suppose that \(h\_i\) is the true hypothesis,
and consider what happens to *each* of its false competitors,
\(h\_j\). If enough evidence becomes available to drive each of the
likelihood ratios 

 

\[\frac{P[e^n \pmid h\_{j}\cdot b\cdot c^{n}]}{P[e^n \pmid h\_{i}\cdot b\cdot c^{n}]}\] 


toward 0 (as *n* increases), then Equation \(9\*\) says that each false
\(h\_j\) will become effectively refuted -- each of their posterior
probabilities will approaches 0 (as *n* increases). As a result, the posterior probability of \(h\_i\) must approach 1. The next two equations show precisely how
this works.



If we sum the ratio versions of Bayes' Theorem in Equation
\(9\*\) over all alternatives to hypothesis \(h\_i\) (including the
catch-all alternative \(h\_K\), if appropriate), we get the Odds Form
of Bayes' Theorem. By definition, the *odds against* a statement \(A\) given \(B\) is related to the probability of \(A\) given \(B\) as follows:

 

 \[\Omega\_{\alpha}[{\nsim}A \pmid B] = \frac{P\_{\alpha}[{\nsim}A \pmid B]}{P\_{\alpha}[A \pmid B]} = \frac{1 - P\_{\alpha}[A \pmid B]}{P\_{\alpha}[A \pmid B]}.\]


This notion of *odds* gives rise to the following version of Bayes' Theorem:




**Bayes' Theorem: Odds Form**

 

\[\tag{10}
\begin{align}
 \Omega\_{\alpha}[{\nsim}h\_i \pmid b\cdot c^{n }\cdot e^{ n}]
& = \sum\_{j\ne i}
\frac{P\_{\alpha}[h\_j \pmid b\cdot c^{n }\cdot e^{ n}]}
{P\_{\alpha}[h\_i \pmid b\cdot c^{n }\cdot e^{ n}]}
\\ &\qquad
+
\frac{P\_{\alpha}[h\_K \pmid b\cdot c^{n }\cdot e^{ n}]}
{P\_{\alpha}[h\_i \pmid b\cdot c^{n }\cdot e^{ n}]}\\[2ex]
& =
\sum\_{ j\ne i}
\frac{P[e^n \pmid h\_{j }\cdot b\cdot c^{ n}]}
{P[e^n \pmid h\_{i }\cdot b\cdot c^{ n}]}
\times
\frac{P\_{\alpha}[h\_j \pmid b]}
{P\_{\alpha}[h\_i \pmid b]}\\[2ex]
&\qquad +
\frac{P\_{\alpha}[e^n \pmid h\_{K }\cdot b\cdot c^{ n}]}
{P[e^n \pmid h\_{i }\cdot b\cdot c^{ n}]}
\times
\frac{P\_{\alpha}[h\_K \pmid b]}
{P\_{\alpha}[h\_i \pmid b]}
\end{align}
\]


where the factor following the '*+*' sign is only
required in cases where a catch-all alternative hypothesis, \(h\_K\),
is needed. 




Recall that when we have a finite collection of concrete alternative
hypotheses available, \(\{h\_1, h\_2 , \ldots ,h\_m\}\), but where this
set of alternatives is not exhaustive (where additional,
unarticulated, undiscovered alternative hypotheses may exist), the
catch-all alternative hypothesis \(h\_K\) is just the denial of each of
the concrete alternatives, \(({\nsim}h\_1\cdot{\nsim}h\_2\cdot \ldots
\cdot{\nsim}h\_m)\). Generally, the likelihood of evidence claims relative to
a catch-all hypothesis will not enjoy the same kind of objectivity possessed by
the likelihoods for concrete alternative hypotheses. So, we leave the
subscript \(\alpha\) attached to the likelihood for the catch-all hypothesis
to indicate this lack of objectivity.



Although the catch-all hypothesis may lack objective likelihoods, the
influence of the catch-all term in Bayes' Theorem diminishes as
additional concrete hypotheses are articulated. That is, as new
hypotheses are discovered they are "peeled off" of the
catch-all. So, when a new hypothesis \(h\_{m+1}\) is formulated and
made explicit, the old catch-all hypothesis \(h\_K\) is replaced by a
new catch-all, \(h\_{K\*}\), of form \(({\nsim}h\_1\cdot
\cdot{\nsim}h\_2\cdot \ldots \cdot{\nsim}h\_{m}\cdot{\nsim}h\_{m+1})\);
and the prior probability for the new catch-all hypothesis is gotten
by diminishing the prior of the old catch-all: \(P\_{\alpha}[h\_{K\*}
\pmid b] = P\_{\alpha}[h\_K \pmid b] - P\_{\alpha}[h\_{m+1} \pmid b]\).
Thus, the influence of the catch-all term should diminish towards 0 as
new alternative hypotheses are made
 explicit.[10]




If increasing evidence drives towards 0 the likelihood ratios
comparing each competitor \(h\_j\) with hypothesis \(h\_i\), then the
odds against \(h\_i\), \(\Omega\_{\alpha}[{\nsim}h\_i \pmid b\cdot
c^{n}\cdot e^{n}]\), will approach 0 (provided that priors of
catch-all terms, if needed, approach 0 as well, as new alternative
hypotheses are made explicit and peeled off). And, as
\(\Omega\_{\alpha}[{\nsim}h\_i \pmid b\cdot c^{n}\cdot e^{n}]\)
approaches 0, the posterior probability of \(h\_i\) goes to 1. This derives from the fact that the odds against \(h\_i\) is related to and its posterior probability by the following formula:




**Bayes' Theorem: General Probabilistic Form**



\[\tag{11}
 P\_{\alpha}[h\_i \pmid b\cdot c^{n}\cdot e^n] = \frac{1}{1 + \Omega\_{\alpha}[{\nsim}h\_i \pmid b\cdot c^n\cdot e^{n}]}.
\]



The odds against a hypothesis depends only on the values of *ratios
of posterior probabilities*, which entirely derive from the Ratio
Form of Bayes' Theorem. Thus, we see that the individual value
of the posterior probability of a hypothesis depends only on the
*ratios of posterior probabilities*, which come from the Ratio
Form of Bayes' Theorem. Thus, the Ratio Form of Bayes'
Theorem captures all the essential features of the Bayesian
evaluation of hypothesis. It shows how the impact of evidence (in the
form of likelihood ratios) combines with comparative plausibility
assessments of hypotheses (in the form of ratios of prior
probabilities) to provide a net assessment of the extent to which
hypotheses are refuted or supported via contests with their rivals.




There is a result, a kind of *Bayesian Convergence Theorem*,
that shows that if \(h\_i\) (together with \(b\cdot c^n)\) is true,
then the likelihood ratios

 

\[\frac{P[e^n \pmid h\_{j}\cdot b\cdot c^{n}]}{P[e^n \pmid h\_{i}\cdot b\cdot c^{n}]}\]


comparing evidentially distinguishable alternative hypothesis \(h\_j\)
to \(h\_i\) will *very probably* approach 0 as evidence
accumulates (i.e., as *n* increases). Let's call this
result the *Likelihood Ratio Convergence Theorem*. When this
theorem applies,
 Equation \(9^\*\)
 shows that the posterior probability of a false competitor \(h\_j\)
will very probably approach 0 as evidence accumulates, regardless of
the value of its prior probability \(P\_{\alpha}[h\_j \pmid b]\). As
this happens to each of \(h\_i\)'s false competitors,
 Equations 10
 and
 11
 say that the posterior probability of the true hypothesis, \(h\_i\),
will approach 1 as evidence
 increases.[11]
 Thus, Bayesian induction is at bottom a version of *induction by
elimination*, where the elimination of alternatives comes by way
of likelihood ratios approaching 0 as evidence accumulates. Thus, when
the *Likelihood Ratio Convergence Theorem* applies, the
*Criterion of Adequacy* for an Inductive Logic described at the
beginning of this article will be satisfied: As evidence accumulates,
the *degree* to which the collection of true evidence
statements comes to *support* a hypothesis, as measured by the
logic, should very probably come to indicate that false hypotheses are
probably false and that true hypotheses are probably true. We will
examine this *Likelihood Ratio Convergence Theorem* in
 Section 4.[12]




A view called *Likelihoodism* relies on likelihood ratios in
much the same way as the Bayesian logic articulated above. However,
*Likelihoodism* attempts to avoid the use of prior
probabilities. For an account of this alternative view, see
the supplement
 Likelihood Ratios, Likelihoodism, and the Law of Likelihood.
For more discussion of
 Bayes' Theorem and its application, see the entries on
 Bayes' Theorem
 and on
 Bayesian Epistemology
 in this *Encyclopedia*.


### 3.4 On Prior Probabilities and Representations of Vague and Diverse Plausibility Assessments



Given that a scientific community should largely agree on the values
of the likelihoods, any significant disagreement among them with
regard to the values of posterior probabilities of hypotheses should
derive from disagreements over their assessments of values for the
prior probabilities of those hypotheses. We saw in
 Section 3.3
 that the Bayesian logic of evidential support need only rely on
assessments of *ratios of prior probabilities*--on how
much more plausible one hypothesis is than another. Thus, the logic of
evidential support only requires that scientists can assess the
comparative plausibilities of various hypotheses. Presumably, in
scientific contexts the comparative plausibility values for hypotheses
should depend on explicit plausibility arguments, not merely on
privately held opinions. (Formally, the logic may represent
comparative plausibility arguments by explicit statements expressed
within \(b\).) It would be highly *unscientific* for a
member of the scientific community to disregard or dismiss a
hypothesis that other members take to be a reasonable proposal with
only the comment, "don't ask me to give my reasons,
it's just my opinion". Even so, agents may be unable to
specify *precisely* how much more strongly the available
plausibility arguments support a hypothesis over an alternative; so
prior probability ratios for hypotheses may be vague. Furthermore,
agents in a scientific community may disagree about how strongly the
available plausibility arguments support a hypothesis over a rival
hypothesis; so prior probability ratios may be somewhat diverse as
well. 



Both the vagueness of comparative plausibilities assessments for
individual agents and the diversity of such assessments among the
community of agents can be represented formally by sets of support
functions, \(\{P\_{\alpha}, P\_{\beta}, \ldots \}\), that agree on the
values for the likelihoods but encompass a range of values for the
(ratios of) prior probabilities of hypotheses. *Vagueness* and
*diversity* are somewhat different issues, but they may be
represented in much the same way. Let's briefly consider each in
turn.



Assessments of the prior plausibilities of hypotheses will often be
vague--not subject to the kind of precise quantitative treatment
that a Bayesian version of probabilistic inductive logic may seem to
require for prior probabilities. So, it may seem that the kind of
assessment of prior probabilities required to get the Bayesian
algorithm going cannot be accomplished in practice. To see how
Bayesian inductivists address this worry, first recall the Ratio Form
of Bayes' Theorem, Equation \(9^\*\).

 

\[
\frac{P\_{\alpha}[h\_j \pmid b\cdot c^{n }\cdot e^{ n}]}
{P\_{\alpha}[h\_i \pmid b\cdot c^{n }\cdot e^{ n}]}
=
\frac{P[e^n \pmid h\_{j }\cdot b\cdot c^{ n}]}
{P[e^n \pmid h\_{i }\cdot b\cdot c^{ n}]}
\times
\frac{P\_{\alpha}[h\_j \pmid b]}
{P\_{\alpha}[h\_i \pmid b]}
\]


Recall that this Ratio Form of the theorem captures the essential
features of the logic of evidential support, even though it only
provides a value for the ratio of the posterior probabilities. Notice
that the ratio form of the theorem easily accommodates situations
where we don't have precise numerical values for prior
probabilities. It only depends on our ability to assess *how much
more or less plausible* alternative hypothesis \(h\_j\) is than
hypothesis \(h\_i\)--only the value of the ratio \(P\_{\alpha}[h\_j
\pmid b] / P\_{\alpha}[h\_i \pmid b]\) need be assessed; the values of
the individual prior probabilities are not needed. Such comparative
plausibilities are much easier to assess than specific numerical
values for the prior probabilities of individual hypotheses. When
combined with the *ratio of likelihoods*, this *ratio of
priors* suffices to yield an assessment of the *ratio of
posterior plausibilities*,

 

 \[\frac{P\_{\alpha}[h\_j \pmid b\cdot c^{n}\cdot e^{n}]}{P\_{\alpha}[h\_i \pmid b\cdot c^{n}\cdot e^{n}]}.\]


Although such posterior ratios don't supply values for the
posterior probabilities of individual hypotheses, they place a crucial
constraint on the posterior support of hypothesis \(h\_j\), since



\[
\begin{align}
P\_{\alpha}[h\_j \pmid b\cdot c^{n }\cdot e^{ n}] 
& \lt 
\frac{P\_{\alpha}[h\_j \pmid b\cdot c^{n }\cdot e^{ n}]}
{P\_{\alpha}[h\_i \pmid b\cdot c^{n }\cdot e^{ n}]}\\
& =
\frac{P[e^n \pmid h\_{j }\cdot b\cdot c^{ n}]}
{P[e^n \pmid h\_{i }\cdot b\cdot c^{ n}]}
\times
\frac{P\_{\alpha}[h\_j \pmid b]}
{P\_{\alpha}[h\_i \pmid b]}
\end{align}
\]


This Ratio Form of Bayes' Theorem tolerates a good deal of
vagueness or imprecision in assessments of the ratios of prior
probabilities. In practice one need only assess bounds for these prior
plausibility ratios to achieve meaningful results. Given a prior ratio
in a specific interval, 

 

\[
q \le \frac{P\_{\alpha}[h\_j \pmid b]}{P\_{\alpha}[h\_i \pmid b]} \le r
\]


a likelihood ratio

 

 \[\frac{P[e^n \pmid h\_{j}\cdot b\cdot c^{n}]}{P[e^n \pmid h\_{i}\cdot b\cdot c^{n}]} = \LR^n\]


results in a posterior support ratio in the interval

 

\[
(\LR^n\times q) \le \frac{P\_{\alpha}[h\_j \pmid b\cdot c^{n}\cdot e^{n}]}{P\_{\alpha}[h\_i \pmid b\cdot c^{n}\cdot e^{n}]} \le (\LR^n \times r).
\]



(Technically each probabilistic support function assigns a specific
numerical value to each pair of sentences; so when we write an
inequality like 

 

 \[q \le \frac{P\_{\alpha}[h\_j \pmid b]}{P\_{\alpha}[h\_i \pmid b]} \le r\]


we are really referring to a set of probability functions
\(P\_{\alpha}\), a *vagueness set*, for which the inequality
holds. Thus, technically, the Bayesian logic employs sets of
probabilistic support functions to represent the vagueness in
comparative plausibility values for hypotheses.)




Observe that if the likelihood ratio values \(\LR^n\) approach 0 as
the amount of evidence \(e^n\) increases, the interval of values for
the posterior probability ratio must become tighter as the upper bound
(\(\LR^n\times r)\) approaches 0. Furthermore, the absolute degree of
support for \(h\_j\), \(P\_{\alpha}[h\_j \pmid b\cdot c^{n}\cdot
e^{n}]\), must also approach 0.



This observation is really useful. For, it can be shown that when
\(h\_{i}\cdot b\cdot c^{n}\) is true and \(h\_j\) is empirically
distinct from \(h\_i\), the continual pursuit of evidence is *very
likely* to result in evidential outcomes \(e^n\) that (as
*n* increases) yield values of likelihood ratios \(P[e^n \pmid
h\_{j}\cdot b\cdot c^{n}] / P[e^n \pmid h\_{i}\cdot b\cdot c^{n}]\) that
approach 0 as the amount of evidence increases. This result, called
the *Likelihood Ratio Convergence Theorem*, will be
investigated in more detail in
 Section 4.
 When that kind of convergence towards 0 for likelihood ratios occurs,
the upper bound on the posterior probability ratio also approaches 0,
driving the posterior probability of \(h\_j\) to approach 0 as well,
effectively refuting hypothesis \(h\_j\). Thus, false competitors of a
true hypothesis will effectively be eliminated by increasing evidence.
As this happens, Equations
 9\*
 through
 11
 show that the posterior probability \(P\_{\alpha}[h\_i \pmid b\cdot
c^{n}\cdot e^{n}]\) of the true hypothesis \(h\_i\) approaches 1. 



Thus, Bayesian logic of inductive support for hypotheses is a form of
eliminative induction, where the evidence effectively refutes false
alternatives to the true hypothesis. Because of its eliminative
nature, the Bayesian logic of evidential support doesn't require
precise values for prior probabilities. It only needs to draw on
bounds on the values of comparative plausibility ratios, and these
bounds only play a significant role while evidence remains fairly
weak. If the true hypothesis is assessed to be comparatively plausible
(due to plausibility arguments contained in *b*), then
plausibility assessments give it a leg-up over alternatives. If the
true hypothesis is assessed to be comparatively implausible, the
plausibility assessments merely slow down the rate at which it comes
to dominate its rivals, reflecting the idea that *extraordinary
hypotheses require extraordinary evidence* (or an extraordinary
accumulation of evidence) to overcome their initial implausibilities.
Thus, as evidence accumulates, the agent's vague initial
plausibility assessments transform into quite sharp posterior
probabilities that indicate their strong refutation or support by the
evidence.



When the various agents in a community may widely disagree over the
non-evidential plausibilities of hypotheses, the Bayesian logic of
evidential support may represent this kind of *diversity*
across the community of agents as a collection of the agents'
*vagueness sets* of support functions. Let's call such a
collection of support functions a *diversity set*. That is, a
*diversity set* is just a set of support functions
\(P\_{\alpha}\) that cover the ranges of values for comparative
plausibility assessments for pairs of competing hypotheses



\[
q \le \frac{P\_{\alpha}[h\_j \pmid b]}{P\_{\alpha}[h\_i \pmid b]} \le r
\]


as assessed by the scientific community. But, once again, if
accumulating evidence drives the likelihood ratios comparing various
alternative hypotheses to the true hypothesis towards 0, the range of
support functions in a *diversity set* will come to near
agreement, near 0, on the values for posterior probabilities of false
competitors of the true hypothesis. So, not only does such evidence
*firm up* each agent's vague initial plausibility
assessment, it also brings the whole community into agreement on the
*near refutation* of empirically distinct competitors of a true
hypothesis. As this happens, the posterior probability of the true
hypothesis may approach 1. The *Likelihood Ratio Convergence
Theorem* implies that this kind of convergence to the truth should
*very probably* happen, provided that the true hypothesis is
empirically distinct enough from its rivals.



One more point about prior probabilities and Bayesian convergence
should be mentioned before proceeding to
 Section 4.
 Some subjectivist versions of Bayesian induction seem to suggest that
an agent's prior plausibility assessments for hypotheses should
stay fixed once-and-for-all, and that all plausibility updating should
be brought about via the likelihoods in accord with Bayes'
Theorem. Critics argue that this is unreasonable. The members of a
scientific community may quite legitimately revise their (comparative)
prior plausibility assessments for hypotheses from time to time as
they rethink plausibility arguments and bring new considerations to
bear. This seems a natural part of the conceptual development of a
science. It turns out that such reassessments of the comparative
plausibilities of hypotheses poses no difficulty for the probabilistic
inductive logic discussed here. Such reassessments may be represented
by the addition or modification of explicit statements that modify the
background information *b*. Such reassessments may result in
(non-Bayesian) transitions to new *vagueness sets* for
individual agents and new *diversity sets* for the community.
The *logic* of Bayesian induction (as described here) has
nothing to say about what values the prior plausibility assessments
for hypotheses should have; and it places no restrictions on how they
might change over time. Provided that the series of reassessments of
(comparative) prior plausibilities doesn't happen to diminish
the (comparative) prior plausibility value of the true hypothesis
towards zero (or, at least, doesn't do so too quickly), the
*Likelihood Ratio Convergence Theorem* implies that the
evidence will very probably bring the posterior probabilities of
empirically distinct rivals of the true hypothesis to approach 0 via
decreasing likelihood ratios; and as this happens, the posterior
probability of the true hypothesis will head towards 1.



(Those interested in a Bayesian account of Enumerative Induction and
the estimation of values for relative frequencies of attributes in
populations should see the supplement,
 Enumerative Inductions: Bayesian Estimation and Convergence.)


## 4. The Likelihood Ratio Convergence Theorem



In this section we will investigate the **Likelihood Ratio
Convergence Theorem**. This theorem shows that under certain
reasonable conditions, when hypothesis \(h\_i\) (in conjunction with
auxiliaries in *b*) is true and an alternative hypothesis \(h\_j\)
is empirically distinct from \(h\_i\) on some possible outcomes of
experiments or observations described by conditions \(c\_k\), then it
is *very likely* that a long enough sequence of such
experiments and observations *c\(^n\)* will produce a sequence
of outcomes \(e^n\) that yields likelihood ratios \(P[e^n \pmid
h\_{j}\cdot b\cdot c^{n}] / P[e^n \pmid h\_{i}\cdot b\cdot c^{n}]\) that
approach 0, favoring \(h\_i\) over \(h\_j\), as evidence accumulates
(i.e., as *n* increases). This theorem places an explicit lower
bound on the "rate of probable convergence" of these
likelihood ratios towards 0. That is, it puts a lower bound on how
likely it is, if \(h\_i\) is true, that a stream of outcomes will occur
that yields likelihood ratio values against \(h\_j\) as compared to
\(h\_i\) that lie within any specified small distance above 0. 



The theorem itself does not require the full apparatus of Bayesian
probability functions. It draws only on likelihoods. Neither the
statement of the theorem nor its proof employ prior probabilities of
any kind. So even *likelihoodists*, who eschew the use of
Bayesian prior probabilities, may embrace this result. Given the forms
of Bayes' Theorem, 9\*-11 from the previous section, the
*Likelihood Ratio Convergence Theorem* further implies the
likely convergence to 0 of the posterior probabilities of false
competitors of a true hypothesis. That is, when the ratios \(P[e^n
\pmid h\_{j}\cdot b\cdot c^{n}] / P[e^n \pmid h\_{i}\cdot b\cdot
c^{n}]\) approach 0 for increasing *n*, the Ratio Form of
Bayes' Theorem,
 Equation 9\*,
 says that the posterior probability of \(h\_j\) must also approach 0
as evidence accumulates, regardless of the value of its prior
probability. So, support functions in collections representing vague
prior plausibilities for an individual agent (i.e., a
*vagueness* set) and representing the diverse range of priors
for a community of agents (i.e., a *diversity* set) will come
to agree on the near 0 posterior probability of empirically distinct
false rivals of a true hypothesis. And as the posterior probabilities
of false competitors fall, the posterior probability of the true
hypothesis heads towards 1. Thus, the theorem establishes that the
inductive logic of probabilistic support functions satisfies the
 Criterion of Adequacy (CoA)
 suggested at the beginning of this article.



The *Likelihood Ratio Convergence Theorem* merely provides some
sufficient conditions for probable convergence. But likelihood ratios
may well converge towards 0 (in the way described by the theorem) even
when the antecedent conditions of the theorem are not satisfied. This
theorem overcomes many of the objections raised by critics of Bayesian
convergence results. First, this theorem does not employ
*second-order probabilities*; it says noting about the
probability of a probability. It only concerns the probability of a
particular disjunctive sentence that expresses a disjunction of
various possible sequences of experimental or observational outcomes.
The theorem does not require evidence to consist of sequences of
events that, according to the hypothesis, are identically distributed
(like repeated tosses of a die). The result is most easily expressed
in cases where the individual outcomes of a sequence of experiments or
observations are probabilistically independent, given each hypothesis.
So that is the version that will be presented in this section.
However, a version of the theorem also holds when the individual
outcomes of the evidence stream are not probabilistically independent,
given the hypotheses. (This more general version of the theorem will
be presented in a supplement on the
 Probabilistic Refutation Theorem,
 below, where the proof of both versions is provided.) In addition,
this result does not rely on supposing that the probability functions
involved are *countably additive*. Furthermore, the explicit
lower bounds on the rate of convergence provided by this result means
that there is no need to wait for the infinitely long run before
convergence occurs (as some critics seem to think).



It is sometimes claimed that Bayesian convergence results only work
when an agent locks in values for the prior probabilities of
hypotheses once-and-for-all, and then updates posterior probabilities
from there only by conditioning on evidence via Bayes Theorem. The
*Likelihood Ratio Convergence Theorem*, however, applies even
if agents revise their prior probability assessments over time. Such
non-Bayesian shifts from one support function (or *vagueness*
set) to another may arise from new plausibility arguments or from
reassessments of the strengths of old ones. The *Likelihood Ratio
Convergence Theorem* itself only involves the values of
likelihoods. So, provided such reassessments don't push the
prior probability of the true hypothesis towards 0 *too
rapidly*, the theorem implies that the posterior probabilities of
each empirically distinct false competitor will *very probably*
approach 0 as evidence
 increases.[13]



### 4.1 The Space of Possible Outcomes of Experiments and Observations



To specify the details of the *Likelihood Ratio Convergence
Theorem* we'll need a few additional notational conventions
and definitions. Here they are.



For a given sequence of *n* experiments or observations \(c^n\),
consider the set of those possible sequences of outcomes that would
result in likelihood ratios for \(h\_j\) over \(h\_i\) that are less
than some chosen small number \(\varepsilon \gt 0\). This set is
represented by the expression,

 

\[
\left\{e^n : \frac{P[e^n \pmid h\_{j }\cdot b\cdot c^{ n}]}{P[e^n \pmid h\_{i }\cdot b\cdot c^{ n}]} \lt \varepsilon \right\}.
\]


Placing the disjunction symbol '\(\vee\)' in front of this
expression yields an expression, 

 

\[
\vee \left\{ e^n : \frac{P[e^n \pmid h\_{j }\cdot b\cdot c^{ n}]}{P[e^n \pmid h\_{i }\cdot b\cdot c^{ n}]} \lt \varepsilon \right\} ,
\]


that we'll use to represent the disjunction of all outcome
sequences \(e^n\) in this set. So,

 

\[
\vee \left\{ e^n : \frac{P[e^n \pmid h\_{j }\cdot b\cdot c^{ n}]}{P[e^n \pmid h\_{i }\cdot b\cdot c^{ n}]} \lt \varepsilon \right\} 
\]


is just a particular sentence that says, in effect, "one of the
sequences of outcomes of the first *n* experiments or
observations will occur that makes the likelihood ratio for \(h\_j\)
over \(h\_i\) less than \(\varepsilon\)".



The *Likelihood Ratio Convergence Theorem* says that under
certain conditions (covered in detail below), the likelihood of a
disjunctive sentence of this sort, given that '\(h\_{i}\cdot
b\cdot c^{n}\)' is true,

 

\[
 P \left[\vee \left\{ e^n : \frac{P[e^n \pmid h\_{j }\cdot b\cdot c^n]}{P[e^n \pmid h\_i\cdot b\cdot c^{ n}]} \lt \varepsilon \right\} \pmid h\_{i }\cdot b\cdot c^{ n}\right] ,
\]


must be at least \(1-(\psi /n)\), for some explicitly calculable term
\(\psi\). Thus, the true hypothesis \(h\_i\) probabilistically implies
that as the amount of evidence, *n*, increases, it becomes highly
likely (as close to 1 as you please) that one of the outcome sequences
\(e^n\) will occur that yields a likelihood ratio \(P[e^n \pmid
h\_{j}\cdot b\cdot c^{n}] / P[e^n \pmid h\_{i}\cdot b\cdot c^{n}]\) less
than \(\varepsilon\); and this holds for any specific value of
\(\varepsilon\) you may choose. As this happens, the posterior
probability of \(h\_i\)'s false competitor, \(h\_j\), must
approach 0, as required by the Ratio Form of Bayes' Theorem,
 Equation 9\*.



The term \(\psi\) in the lower bound of this probability depends on a
measure of the empirical distinctness of the two hypotheses \(h\_j\)
and \(h\_i\) for the proposed sequence of experiments and observations
\(c^n\). To specify this measure we need to contemplate the collection
of possible outcomes of each experiment or observation. So, consider
some sequence of experimental or observational conditions described by
sentences \(c\_1,c\_2 ,\ldots ,c\_n\). Corresponding to each condition
\(c\_k\) there will be some range of possible alternative outcomes. Let
\(O\_{k} = \{o\_{k1},o\_{k2},\ldots ,o\_{kw}\}\) be a set of statements
describing the alternative possible outcomes for condition \(c\_k\).
(The number of alternative outcomes will usually differ for distinct
experiments among those in the sequence \(c\_1 ,\ldots ,c\_n\); so, the
value of *w* may depend on \(c\_k\).) For each hypothesis \(h\_j\),
the alternative outcomes of \(c\_k\) in \(O\_k\) are mutually exclusive
and exhaustive, so we have:

 

\[
 P[o\_{ku }\cdot o\_{kv} \pmid h\_j\cdot b\cdot c\_{ k}] = 0 
\textrm{ and }
\sum^{w}\_{u = 1} P[o\_{ku} \pmid h\_{j }\cdot b\cdot c\_{ k}] =1 . 
\]


We now let expressions of form '\(e\_k\)' act as variables
that range over the possible outcomes of condition \(c\_k\)--i.e.,
\(e\_k\) ranges over the members of \(O\_k\). As before,
'\(c^n\)' denotes the conjunction of the first *n*
test conditions, \((c\_1\cdot c\_2\cdot \ldots \cdot c\_n)\), and
'\(e^n\)' represents possible sequences of corresponding
outcomes, \((e\_1\cdot e\_2\cdot \ldots \cdot e\_n)\). Let's use
the expression '*E\(^n\)*' to represent the set of
all possible outcome sequences that may result from the sequence of
conditions *c\(^n\)*. So, for each hypothesis \(h\_j\)
(including \(h\_i)\), \(\sum\_{e^n\in E^n} P[e^n \pmid h\_{j}\cdot b\cdot
c^{n}] = 1\). 



Everything introduced in this subsection is mere notational
convention. No substantive suppositions (other than the axioms of
probability theory) have yet been introduced. The version of the
*Likelihood Ratio Convergence Theorem* I'll present below
does, however, draw on one substantive supposition, although a rather
weak one. The next subsection will discuss that supposition in
detail.


### 4.2 Probabilistic Independence



In most scientific contexts the outcomes in a stream of experiments or
observations are *probabilistically independent* of one another
relative to each hypothesis under consideration, or can at least be
divided up into probabilistically independent parts. For our purposes
*probabilistic independence of evidential outcomes on a
hypothesis* divides neatly into two types.




**Definition: Independent Evidence Conditions**:


1. A sequence of outcomes \(e^k\) is
**condition-independent** of a condition for an
additional experiment or observation \(c\_{k+1}\), given \(h\cdot b\)
together with its own conditions \(c^k\), *if and only if*

\[
P[e^k \pmid h\cdot b\cdot c^{k }\cdot c\_{ k+1}] = P[e^k \pmid h\cdot b\cdot c^k] .
\]
2. An individual outcome \(e\_k\) is **result-independent** of a sequence of other observations and
their outcomes \((c^{k-1}\cdot e^{k-1})\), given \(h\cdot b\) and its
own condition \(c\_k\), *if and only if* 

\[
P[e\_k \pmid h\cdot b\cdot c\_k\cdot(c^{k-1 }\cdot e^{ k-1})] = P[e\_k \pmid h\cdot b\cdot c\_k] .
 \]




When these two conditions hold, the likelihood for an evidence
sequence may be decomposed into the product of the likelihoods for
individual experiments or observations. To see how the two
*independence conditions* affect the decomposition, first
consider the following formula, which holds even when neither
*independence condition* is satisfied:

 

\[\tag{12}
 P[e^n \pmid h\_{j }\cdot b\cdot c^{ n}] = 
\prod^{n}\_{k = 1}
 P[e\_k \pmid h\_{j }\cdot b\cdot c^n\cdot e^{ k-1}] . 
\]


When *condition-independence* holds, the likelihood of the
whole evidence stream parses into a product of likelihoods that
*probabilistically depend* on only past observation conditions
and their outcomes. They do not depend on the conditions for other
experiments whose outcomes are not yet specified. Here is the
formula:

 

\[\tag{13}
 P[e^n \pmid h\_{j }\cdot b\cdot c^{ n}] = 
 \prod^{n}\_{k = 1}
 P[e\_k \pmid h\_{j }\cdot b\cdot c\_k\cdot (c^{k-1}\cdot e^{ k-1})] . 
\]


Finally, whenever both *independence conditions* are satisfied
we have the following relationship between the likelihood of the
evidence stream and the likelihoods of individual experiments or
observations:

 

\[\tag{14}
 P[e^n \pmid h\_{j }\cdot b\cdot c^{ n}] = 
\prod^{n}\_{k = 1}
 P[e\_k \pmid h\_{j }\cdot b\cdot c\_{ k}] . 
\]


(For proofs of Equations 12-14 see the supplement
 Immediate Consequences of Independent Evidence Conditions.)



In scientific contexts the evidence can almost always be divided into
parts that satisfy both clauses of the *Independent Evidence
Condition* with respect to each alternative hypothesis. To see
why, let us consider each independence condition more carefully.



*Condition-independence* says that the mere addition of a new
observation condition \(c\_{k+1}\), *without specifying one of its
outcomes*, does not alter the likelihood of the outcomes \(e^k\)
of other experiments \(c^k\). To appreciate the significance of this
condition, imagine what it would be like if it were violated. Suppose
hypothesis \(h\_j\) is some statistical theory, say, for example, a
quantum theory of superconductivity. The conditions expressed in
\(c^k\) describe a number of experimental setups, perhaps conducted in
numerous labs throughout the world, that test a variety of aspects of
the theory (e.g., experiments that test electrical conductivity in
different materials at a range of temperatures). An outcome sequence
\(e^k\) describes the results of these experiments. The violation of
*condition-independence* would mean that merely adding to
\(h\_{j}\cdot b\cdot c^{k}\) a statement \(c\_{k+1}\) describing how an
additional experiment has been set up, but with no mention of its
outcome, changes how likely the evidence sequence \(e^k\) is taken to
be. What \((h\_j\cdot b)\) *says* via likelihoods about the
outcomes \(e^k\) of experiments \(c^k\) differs as a result of merely
supplying a description of another experimental arrangement,
\(c\_{k+1}\). *Condition-independence*, when it holds, rules out
such strange effects.



*Result-independence* says that the description of previous
test conditions *together with their outcomes* is irrelevant to
the likelihoods of outcomes for additional experiments. If this
condition were widely violated, then in order to specify the most
informed likelihoods for a given hypothesis one would need to include
information about volumes of past observations and their outcomes.
What a hypothesis says about future cases would depend on how past
cases have gone. Such *dependence* had better not happen on a
large scale. Otherwise, the hypothesis would be fairly useless, since
its empirical import in each specific case would depend on taking into
account volumes of past observational and experimental results.
However, even if such dependencies occur, provided they are not too
pervasive, *result-independence* can be accommodated rather
easily by packaging each collection of *result-dependent* data
together, treating it like a single extended experiment or
observation. The *result-independence condition* will then be
satisfied by letting each term '\(c\_k\)' in the statement
of the independence condition represent a conjunction of test
conditions for a collection of *result-dependent* tests, and by
letting each term '\(e\_k\)' (and each term
'\(o\_{ku}\)') stand for a conjunction of the corresponding
*result-dependent* outcomes. Thus, by packaging
*result-dependent* data together in this way, the
*result-independence* condition is satisfied by those
(conjunctive) statements that describe the separate,
*result-independent*
 chunks.[14]




The version of the *Likelihood Ratio Convergence Theorem* we
will examine depends only on the *Independent Evidence
Conditions* (together with the axioms of probability theory). It
draws on no other assumptions. Indeed, an even more general version of
the theorem can be established, a version that draws on neither of the
*Independent Evidence Conditions*. However, the *Independent
Evidence Conditions* will be satisfied in almost all scientific
contexts, so little will be lost by assuming them. (And the
presentation will run more smoothly if we side-step the added
complications needed to explain the more general result.)



From this point on, let us assume that the following versions of the
*Independent Evidence Conditions* hold. 




**Assumption: Independent Evidence Assumptions**. For
each hypothesis *h* and background *b* under consideration,
we assume that the experiments and observations can be packaged into
condition statements, \(c\_1 ,\ldots ,c\_k, c\_{k+1},\ldots\), and
possible outcomes in a way that satisfies the following
conditions:


1. Each sequence of possible outcomes \(e^k\) of a sequence of
conditions \(c^k\) is **condition-independent** of
additional conditions \(c\_{k+1}\)--i.e., 

\[P[e^k \pmid h\cdot b\cdot c^{k}\cdot c\_{k+1}] = P[e^k \pmid h\cdot b\cdot c^k].\]
2. Each possible outcome \(e\_k\) of condition \(c\_k\) is
**result-independent** of sequences of other observations
and possible outcomes \((c^{k-1}\cdot e^{k-1})\)--i.e.,

\[P[e\_k \pmid h\cdot b\cdot c\_k\cdot(c^{k-1}\cdot e^{k-1})] = P[e\_k \pmid h\cdot b\cdot c\_k].\]




We now have all that is needed to begin to state the *Likelihood
Ratio Convergence Theorem*.


### 4.3 Likelihood Ratio Convergence when Falsifying Outcomes are Possible



The *Likelihood Ratio Convergence Theorem* comes in two parts.
The first part applies only to those experiments or observations
\(c\_k\) within the total evidence stream \(c^n\) for which some of the
possible outcomes have 0 likelihood of occurring according to
hypothesis \(h\_j\) but have non-0 likelihood of occurring according to
\(h\_i\). Such outcomes are highly desirable. If they occur, the
likelihood ratio comparing \(h\_j\) to \(h\_i\) will become 0, and
\(h\_j\) will be *falsified*. So-called *crucial
experiments* are a special case of this, where for at least one
possible outcome \(o\_{ku}\), \(P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}]
= 1\) and \(P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}] = 0\). In the more
general case \(h\_i\) together with *b* says that one of the
outcomes of \(c\_k\) is at least minimally probable, whereas \(h\_j\)
says that this outcome is impossible--i.e., \(P[o\_{ku} \pmid
h\_{i}\cdot b\cdot c\_{k}] \gt 0\) and \(P[o\_{ku} \pmid h\_{j}\cdot
b\cdot c\_{k}] = 0\). It will be convenient to define a term for this
situation.




**Definition: Full Outcome Compatibility.** Let's
call \(h\_j\) *fully outcome-compatible* with \(h\_i\) on
experiment or observation \(c\_k\) *just when*, for each of its
possible outcomes \(e\_k\), if \(P[e\_k \pmid h\_{i}\cdot b\cdot c\_{k}]
\gt 0\), then \(P[e\_k \pmid h\_{j}\cdot b\cdot c\_{k}] \gt 0\).
Equivalently, \(h\_j\) is *fails to be fully outcome-compatible*
with \(h\_i\) on experiment or observation \(c\_k\) *just when*,
for at least one of its possible outcomes \(e\_k\), \(P[e\_k \pmid
h\_{i}\cdot b\cdot c\_{k}] \gt 0\) but \(P[e\_k \pmid h\_{j}\cdot b\cdot
c\_{k}] = 0\). 




The first part of the *Likelihood Ratio Convergence Theorem*
applies to that part of the total stream of evidence (i.e., that
subsequence of the total evidence stream) on which hypothesis \(h\_j\)
*fails to be fully outcome-compatible* with hypothesis \(h\_i\);
the second part of the theorem applies to the remaining part of the
total stream of evidence, that subsequence of the total evidence
stream on which \(h\_j\) is *fully outcome-compatible* with
\(h\_i\). It turns out that these two kinds of cases must be treated
differently. (This is due to the way in which the *expected
information content* for empirically distinguishing between the
two hypotheses will be measured for experiments and observations that
are *fully outcome compatible*; this measure of information
content blows up (becomes infinite) for experiments and observations
that *fail to be fully outcome compatible*). Thus, the
following part of the convergence theorem applies to just that part of
the total stream of evidence that consists of experiments and
observations that *fail to be fully outcome compatible* for the
pair of hypotheses involved. Here, then, is the first part of the
convergence theorem.




**Likelihood Ratio Convergence Theorem 1--The Falsification
Theorem:**
  

Suppose that the total stream of evidence \(c^n\) contains precisely
*m* experiments or observations on which \(h\_j\) *fails to be
fully outcome-compatible* with \(h\_i\). And suppose that the
*Independent Evidence Conditions* hold for evidence stream
\(c^n\) with respect to each of these two hypotheses. Furthermore,
suppose there is a lower bound \(\delta \gt 0\) such that for each
\(c\_k\) on which \(h\_j\) *fails to be fully outcome-compatible*
with \(h\_i\), 

 

\[P[\vee \{ o\_{ku} : P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}] = 0\} \pmid h\_{i}\cdot b\cdot c\_{k}] \ge \delta\]


--i.e., \(h\_i\) together with \(b\cdot c\_k\) *says*, with
likelihood at least as large as \(\delta\), that one of the outcomes
will occur that \(h\_j\) *says* cannot occur. Then, 



\[\begin{align}
P \left[\vee \left\{ e^n : \frac{P[e^n \pmid h\_{j}\cdot b\cdot c^n]}{P[e^n \pmid h\_i\cdot b\cdot c^{n}]} = 0\right\} \pmid h\_{i}\cdot b\cdot c^{n}\right]\\[2ex]
 = P\left[\vee \left\{ e^n : P[e^n \pmid h\_{j}\cdot b\cdot c^{n}] = 0\right\} \pmid h\_{i}\cdot b\cdot c^{n}\right]\\
 \ge 1 - (1-\delta)^m,
\end{align}
\]


which approaches 1 for large *m*. (For proof see
 Proof of the Falsification Theorem.)
 




In other words, we only suppose that for each of *m*
observations, \(c\_k, h\_i\) *says* observation \(c\_k\) has at
least a small likelihood \(\delta\) of producing one of the outcomes
\(o\_{ku}\) that \(h\_j\) *says* is impossible. If the number
*m* of such experiments or observations is large enough (or if
the lower bound \(\delta\) on the likelihoods of getting such outcomes
is large enough), and if \(h\_i\) (together with \(b\cdot c^n)\) is
true, then it is highly likely that one of the outcomes held to be
impossible by \(h\_j\) will actually occur. If one of these outcomes
does occur, then the likelihood ratio for \(h\_j\) as compared to over
\(h\_i\) will become 0. According to Bayes' Theorem, when this
happen, \(h\_j\) is absolutely refuted by the evidence--its
posterior probability becomes 0.



The Falsification Theorem is quite commonsensical. First, notice that
if there is a *crucial experiment* in the evidence stream, the
theorem is completely obvious. That is, suppose for the specific
experiment \(c\_k\) (in evidence stream \(c^n)\) there are two
incompatible possible outcomes \(o\_{kv}\) and \(o\_{ku}\) such that
\(P[o\_{kv} \pmid h\_{j}\cdot b\cdot c\_{k}] = 1\) and \(P[o\_{ku} \pmid
h\_{i}\cdot b\cdot c\_{k}] = 1\). Then, clearly, \(P[\vee \{ o\_{ku}:
P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}] = 0\} \pmid h\_{i}\cdot b\cdot
c\_{k}] = 1\), since \(o\_{ku}\) is one of the \(o\_{ku}\) such that
\(P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}] = 0\). So, where a crucial
experiment is available, the theorem applies with \(m = 1\) and
\(\delta = 1\).



The theorem is equally commonsensical for cases where no crucial
experiment is available. To see what it says in such cases, consider
an example. Let \(h\_i\) be some theory that implies a specific rate of
proton decay, but a rate so low that there is only a very small
probability that any particular proton will decay in a given year.
Consider an alternative theory \(h\_j\) that implies that protons
*never* decay. If \(h\_i\) is true, then for a persistent enough
sequence of observations (i.e., if proper detectors can keep trillions
of protons under observation for long enough), eventually a proton
decay will almost surely be detected. When this happens, the
likelihood ratio becomes 0. Thus, the posterior probability of \(h\_j\)
becomes 0.



It is instructive to plug some specific values into the formula given
by the Falsification Theorem, to see what the convergence rate might
look like. For example, the theorem tells us that if we compare any
pair of hypotheses \(h\_i\) and \(h\_j\) on an evidence stream \(c^n\)
that contains at least \(m = 19\) observations or experiments, where
each has a likelihood \(\delta \ge .10\) of yielding a *falsifying
outcome*, then the likelihood (on \(h\_{i}\cdot b\cdot c^{n})\) of
obtaining an outcome sequence \(e^n\) that yields likelihood-ratio



 \[\frac{P[e^n \pmid h\_{j}\cdot b\cdot c^{n}] }{P[e^n \pmid h\_{i}\cdot b\cdot c^{n}]} = 0,\]


will be at least as large as \((1 - (1-.1)^{19}) = .865\). (The reader
is invited to try other values of \(\delta\) and *m*.)



A comment about the *need for* and *usefulness of* such
convergence theorems is in order, now that we've seen one. Given
some specific pair of scientific hypotheses \(h\_i\) and \(h\_j\) one
may directly compute the likelihood, given \((h\_{i}\cdot b\cdot
c^{n})\), that a proposed sequence of experiments or observations
\(c^n\) will result in one of the sequences of outcomes that would
yield low likelihood ratios. So, given a specific pair of hypotheses
and a proposed sequence of experiments, we don't need a general
*Convergence Theorem* to tell us the likelihood of obtaining
refuting evidence. The specific hypotheses \(h\_i\) and \(h\_j\) tell us
this *themselves*. They tell us the likelihood of obtaining
each specific outcome stream, including those that either refute the
competitor or produce a very small likelihood ratio for it.
Furthermore, after we've actually performed an experiment and
recorded its outcome, all that matters is the actual ratio of
likelihoods for that outcome. Convergence theorems become moot.



The point of the Likelihood Ratio Convergence Theorem (both the
Falsification Theorem and the part of the theorem still to come) is to
assure us *in advance of considering any specific pair of
hypotheses* that if the possible evidence streams that test
hypotheses have certain characteristics which reflect the empirical
distinctness of the two hypotheses, then it is highly likely that one
of the sequences of outcomes will occur that yields a very small
likelihood ratio. These theorems provide finite lower bounds on how
quickly such convergence is likely to be. Thus, they show that the
 CoA
 is satisfied in advance of our using the logic to test specific pairs
of hypotheses against one another.


### 4.4 Likelihood Ratio Convergence When No Falsifying Outcomes are Possible



The Falsification Theorem applies whenever the evidence stream
includes possible outcomes that may *falsify* the alternative
hypothesis. However, it completely ignores the influence of any
experiments or observations in the evidence stream on which hypothesis
\(h\_j\) is *fully outcome-compatible* with hypothesis \(h\_i\).
We now turn to a theorem that applies to those evidence streams (or to
parts of evidence streams) consisting only of experiments and
observations on which hypothesis \(h\_j\) is *fully
outcome-compatible* with hypothesis \(h\_i\). Evidence streams of
this kind contain no *possibly falsifying* outcomes. In such
cases the only outcomes of an experiment or observation \(c\_k\) for
which hypothesis \(h\_j\) may specify 0 likelihoods are those for which
hypothesis \(h\_i\) specifies 0 likelihoods as well. 



Hypotheses whose connection with the evidence is entirely statistical
in nature will usually be *fully outcome-compatible* on the
entire evidence stream. So, evidence streams of this kind are
undoubtedly much more common in practice than those containing
possibly falsifying outcomes. Furthermore, whenever an entire stream
of evidence contains some mixture of experiments and observations on
which the hypotheses are *not fully outcome compatible* along
with others on which they are *fully outcome compatible*, we
may treat the experiments and observations for which *full outcome
compatibility* holds as a separate subsequence of the entire
evidence stream, to see the likely impact of that part of the evidence
in producing values for likelihood ratios. 



To cover evidence streams (or subsequences of evidence streams)
consisting entirely of experiments or observations on which \(h\_j\) is
*fully outcome-compatible* with hypothesis \(h\_i\) we will
first need to identify a useful way to measure the degree to which
hypotheses are empirically distinct from one another on such evidence.
Consider some particular sequence of outcomes \(e^n\) that results
from observations \(c^n\). The likelihood ratio \(P[e^n \pmid
h\_{j}\cdot b\cdot c^{n}] / P[e^n \pmid h\_{i}\cdot b\cdot c^{n}]\)
itself measures the extent to which the outcome sequence distinguishes
between \(h\_i\) and \(h\_j\). But as a measure of the power of evidence
to distinguish among hypotheses, raw likelihood ratios provide a
rather lopsided scale, a scale that ranges from 0 to infinity with the
midpoint, where \(e^n\) doesn't distinguish at all between
\(h\_i\) and \(h\_j\), at 1. So, rather than using raw likelihood ratios
to measure the ability of \(e^n\) to distinguish between hypotheses,
it proves more useful to employ a symmetric measure. The logarithm of
the likelihood ratio provides such a measure.




**Definition: QI--the Quality of the Information**.
  

For each experiment or observation \(c\_k\), define *the quality of
the information* provided by possible outcome \(o\_{ku}\) for
distinguishing \(h\_j\) from \(h\_i\), given *b*, as follows (where
henceforth we take "logs" to be base-2):

 

\[
\QI[o\_{ku} \pmid h\_i /h\_j \pmid b\cdot c\_k] = \log\left[\frac{P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_k]}{P[o\_{ku} \pmid h\_j\cdot b\cdot c\_{k}]}\right].
\]


Similarly, for the sequence of experiments or observations \(c^n\),
define *the quality of the information* provided by possible
outcome \(e^n\) for distinguishing \(h\_j\) from \(h\_i\), given
*b*, as follows:

 

\[
\QI[e^n \pmid h\_i /h\_j \pmid b\cdot c^n] = \log\left[\frac{P[e^n \pmid h\_{i}\cdot b\cdot c^n]}{P[e^n \pmid h\_j\cdot b\cdot c^{n}]}\right].
\]


That is, QI is the base-2 logarithm of the likelihood ratio for
\(h\_i\) over that for \(h\_j\).




So, we'll measure the *Quality of the Information* an
outcome would yield in distinguishing between two hypotheses as the
base-2 logarithm of the likelihood ratio. This is clearly a symmetric
measure of the outcome's evidential strength at distinguishing
between the two hypotheses. On this measure hypotheses \(h\_i\) and
\(h\_j\) assign the same likelihood value to a given outcome \(o\_{ku}\)
*just when* \(\QI[o\_{ku} \pmid h\_i /h\_j \pmid b\cdot c\_k] =
0\). Thus, QI measures information on a logarithmic scale that is
symmetric about the natural no-information midpoint, 0. This measure
is set up so that *positive information* favors \(h\_i\) over
\(h\_j\), and *negative information* favors \(h\_j\) over
\(h\_i\).



Given the *Independent Evidence Assumptions* with respect to
each hypothesis, it's easy to show that the QI for a sequence of
outcomes is just the sum of the QIs of the individual outcomes in the
sequence:

 

\[\tag{15}
 \QI[e^n \pmid h\_i /h\_j \pmid b\cdot c^n] =
\sum^{n}\_{k = 1}
 \QI[e\_k \pmid h\_i /h\_j \pmid b\cdot c\_k]. 
\]


Probability theorists measure the *expected value* of a
quantity by first multiplying each of its *possible values* by
their probabilities of occurring, and then summing these products.
Thus, the *expected value* of QI is given by the following
formula:




**Definition: EQI--the Expected Quality of the
Information**.
  

We adopt the convention that if \(P[o\_{ku} \pmid h\_{i}\cdot b\cdot
c\_{k}] = 0\), then the term \(\QI[o\_{ku} \pmid h\_i /h\_j \pmid b\cdot
c\_k] \times P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}] = 0\). This
convention will make good sense in the context of the following
definition because, whenever the outcome \(o\_{ku}\) has 0 probability
of occurring according to \(h\_i\) (together with \(b\cdot c\_k)\), it
makes good sense to give it 0 impact on the ability of the evidence to
distinguish between \(h\_j\) and \(h\_i\) when \(h\_i\) (together with
\(b\cdot c\_k)\) is true. Also notice that the *full
outcome-compatibility* of \(h\_j\) with \(h\_i\) on \(c\_k\) means
that whenever \(P[e\_k \pmid h\_{j}\cdot b\cdot c\_{k}] = 0\), we must
have \(P[e\_k \pmid h\_{i}\cdot b\cdot c\_{k}] = 0\) as well; so whenever
the denominator would be 0 in the term 

 

\[\QI[o\_{ku} \pmid h\_i /h\_j \pmid b\cdot c\_k] = \log\left[\frac{P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_k]}{P[o\_{ku} \pmid h\_j\cdot b\cdot c\_{k}]}\right],\]


the convention just described makes the term 

 

\[\QI[o\_{ku} \pmid h\_i /h\_j \pmid b\cdot c\_k] \times P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}] = 0.\]


Thus the following notion is well-defined: 



For \(h\_j\) *fully outcome-compatible* with \(h\_i\) on
experiment or observation \(c\_k\), define

 

\[
\EQI[c\_k \pmid h\_i /h\_j \pmid b] = \sum\_u \QI[o\_{ku} \pmid h\_i /h\_j \pmid b\cdot c\_k] \times P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}].
\]


Also, for \(h\_j\) *fully outcome-compatible* with \(h\_i\) on
each experiment and observation in the sequence \(c^n\), define



 \[
\EQI[c^n \pmid h\_i /h\_j \pmid b]
 =
 \sum\_{e^n\in E^n}
 \QI[e^n \pmid h\_i /h\_j \pmid b\cdot c^n] \times P[e^n \pmid h\_{i}\cdot b\cdot c^{n}]. 
\]



The EQI of an experiment or observation is the *Expected Quality of
its Information* for distinguishing \(h\_i\) from \(h\_j\) when
\(h\_i\) is true. It is a measure of the expected evidential strength
of the possible outcomes of an experiment or observation at
distinguishing between the hypotheses when \(h\_i\) (together with
\(b\cdot c)\) is true. Whereas QI measures the ability of each
particular outcome or sequence of outcomes to empirically distinguish
hypotheses, EQI measures the tendency of experiments or observations
to produce distinguishing outcomes. It can be shown that EQI tracks
empirical distinctness in a very precise way. We return to this in a
moment.



It is easily seen that the EQI for a sequence of observations \(c^n\)
is just the sum of the EQIs of the individual observations \(c\_k\) in
the sequence:

 

\[\tag{16}
 \EQI[c^n \pmid h\_i /h\_j \pmid b]
 =
 \sum^{n}\_{k=1}
 \EQI[c\_k \pmid h\_i /h\_j \pmid b\_{}]. 
\]


(For proof see the supplement
 Proof that the EQI for \(c^n\) is the sum of the EQI for the individual \(c\_k\).)



This suggests that it may be useful to average the values of the
\(\EQI[c\_k \pmid h\_i /h\_j \pmid b]\) over the number of observations
*n* to obtain a measure of the *average expected quality of
the information* among the experiments and observations that make
up the evidence stream \(c^n\).




**Definition: The Average Expected Quality of
Information**
  

For \(h\_j\) *fully outcome-compatible* with \(h\_i\) on each
experiment and observation in the evidence stream \(c^n\), define the
average expected quality of information, \(\bEQI\), from \(c^n\) for
distinguishing \(h\_j\) from \(h\_i\), given \(h\_i\cdot b\), as
follows:

 

\[
\begin{align}
 \bEQI[c^n \pmid h\_i /h\_j \pmid b] 
& =
 \frac{\EQI[c^n \pmid h\_i /h\_j \pmid b]}{n}\\
& =
 (1/n) \times
 \sum^{n}\_{k=1}
 \EQI[c\_k \pmid h\_i /h\_j \pmid b\_{}].
\end{align}
 \]



It turns out that the value of \(\EQI[c\_k \pmid h\_i /h\_j \pmid b\_{}]\)
cannot be less than 0; and it must be greater than 0 just in case
\(h\_i\) is *empirically distinct* from \(h\_j\) on at least one
outcome \(o\_{ku}\)--i.e., just in case it is *empirically
distinct* in the sense that \(P[o\_{ku} \pmid h\_{i}\cdot b\cdot
c\_{k}] \ne P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}]\), for at least one
outcome \(o\_{ku}\). The same goes for the average, \(\bEQI[c^n \pmid
h\_i /h\_j \pmid b]\).




**Theorem: Nonnegativity of EQI.**



\(\EQI[c\_k \pmid h\_i /h\_j \pmid b\_{}] \ge 0\); and \(\EQI[c\_k \pmid
h\_i /h\_j \pmid b\_{}] \gt 0\) *if and only if* for at least one
of its possible outcomes \(o\_{ku}\), 

 

\[P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}] \ne P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}].\]


As a result, \(\bEQI[c^n \pmid h\_i /h\_j \pmid b] \ge 0\); and
\(\bEQI[c^n \pmid h\_i /h\_j \pmid b] \gt 0\) *if and only if* at
least one experiment or observation \(c\_k\) has at least one possible
outcome \(o\_{ku}\) such that 

 

\[P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}] \ne P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}].\]


(For proof, see the supplement
 The Effect on EQI of Partitioning the Outcome Space More Finely--Including Proof of the Nonnegativity of EQI.)




In fact, the more finely one partitions the outcome space \(O\_{k} =
\{o\_{k1},\ldots ,o\_{kv},\ldots ,o\_{kw}\}\) into distinct outcomes that
differ on likelihood ratio values, the larger EQI
 becomes.[15]
 This shows that EQI tracks empirical distinctness in a precise way.
The importance of the *Non-negativity of EQI* result for the
*Likelihood Ratio Convergence Theorem* will become clear in a
moment.



We are now in a position to state the second part of the
*Likelihood Ratio Convergence Theorem*. It applies to all
evidence streams not containing *possibly falsifying outcomes*
for \(h\_j\) when \(h\_i\) holds--i.e., it applies to all evidence
streams for which \(h\_j\) is *fully outcome-compatible* with
\(h\_i\) on each \(c\_k\) in the stream.




**Likelihood Ratio Convergence Theorem 2--The Probabilistic
Refutation Theorem.**



Suppose the evidence stream \(c^n\) contains only experiments or
observations on which \(h\_j\) is *fully outcome-compatible*
with \(h\_i\)--i.e., suppose that for each condition \(c\_k\) in
sequence \(c^n\), for each of its possible outcomes possible outcomes
\(o\_{ku}\), either \(P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}] = 0\) or
\(P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}] \gt 0\). In addition (as a
slight strengthening of the previous supposition), for some \(\gamma
\gt 0\) a number smaller than \(1/e^2\) (\(\approx .135\); where
*e*' is the base of the natural logarithm), suppose that
for each possible outcome \(o\_{ku}\) of each observation condition
\(c\_k\) in \(c^n\), either \(P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}] =
0\) or 

 

\[\frac{P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}]}{P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}]} \ge \gamma.\]


And suppose that the *Independent Evidence Conditions* hold for
evidence stream \(c^n\) with respect to each of these hypotheses. Now,
choose any positive \(\varepsilon \lt 1\), as small as you like, but
large enough (for the number of observations *n* being
contemplated) that the value of 

 

 \[\bEQI[c^n \pmid h\_i /h\_j \pmid b] \gt -\frac{(\log \varepsilon)}{n}.\]


Then:

 

\[
\begin{multline}
P\left[\vee \left\{ e^n : \frac{P[e^n \pmid h\_{j}\cdot b\cdot c^{n}]}{P[e^n \pmid h\_{i}\cdot b\cdot c^{n}]} \lt \varepsilon \right\} \pmid h\_{i}\cdot b\cdot c^{n}\right]\\[2ex]
\gt 1 - \frac{1}{n} \times 
\frac{(\log \gamma)^2}
{(\bEQI[c^n \pmid h\_i /h\_j \pmid b] + (\log \varepsilon)/n)^2}
\end{multline}
\]


For \(\varepsilon = 1/2^m\) and \(\gamma = 1/2^q\), this formula
becomes,

 

\[
\begin{multline}
P\left[\vee \left\{ e^n : \frac{P[e^n \pmid h\_{j}\cdot b\cdot c^{n}]}{P[e^n \pmid h\_{i}\cdot b\cdot c^{n}]} \lt 1/2^m\right\} \pmid h\_{i}\cdot b\cdot c^{n}\right]\\
 \gt 1 - \frac{1}{n}
 \times
\frac{q^2}
{(\bEQI[c^n \pmid h\_i /h\_j \pmid b] - (m/n) )^2}
 \end{multline}
\]


(For proof see the supplement
 Proof of the Probabilistic Refutation Theorem.)
 




This theorem provides sufficient conditions for the *likely
refutation* of false alternatives via exceeding small likelihood
ratios. The conditions under which this happens characterize the
degree to which the hypotheses involved are empirically distinct from
one another. The theorem says that when these conditions are met,
according to hypothesis \(h\_i\) (taken together with \(b\cdot c^n)\),
the likelihood is near 1 that that one of the outcome sequence \(e^n\)
will occur for which the likelihood ratio is smaller than
\(\varepsilon\) (for any value of \(\varepsilon\) you may choose). The
likelihood of getting such an evidential outcome \(e^n\) is quite
close to 1--i.e., no more than the amount 

 

\[\frac{1}{n} \times \frac{(\log \gamma)^2}{\left(\bEQI[c^n \pmid h\_i /h\_j \pmid b] + \frac{(\log \varepsilon)}{n}\right)^2}\] 


below 1. (Notice that this amount below 1 goes to 0 as *n*
increases.) 



It turns out that in almost every case (for almost any pair of
hypotheses) the actual likelihood of obtaining such evidence (i.e.,
evidence that has a likelihood ratio value less than \(\varepsilon)\)
will be *much closer* to 1 than this factor
 indicates.[16]
 Thus, the theorem provides an overly cautious lower bound on the
likelihood of obtaining small likelihood ratios. It shows that the
larger the value of \(\bEQI\) for an evidence stream, the more likely
that stream is to produce a sequence of outcomes that yield a very
small likelihood ratio value. But even if \(\bEQI\) remains quite
small, a long enough evidence stream, *n*, of such low-grade
evidence will, nevertheless, almost surely produce an outcome sequence
having a very small likelihood ratio
 value.[17]




Notice that the antecedent condition of the theorem, that
"either

 

\[P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}] = 0\] 


or

 

\[\frac{P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}]}{P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}]} \ge \gamma,\]


for some \(\gamma \gt 0\) but less than \(1/e^2\) (\(\approx
.135\))", does not favor hypothesis \(h\_i\) over \(h\_j\) in any
way. The condition only rules out the possibility that some outcomes
might furnish *extremely strong* evidence *against*
\(h\_j\) relative to \(h\_i\)--by making \(P[o\_{ku} \pmid
h\_{i}\cdot b\cdot c\_{k}] = 0\) or by making 

 

\[\frac{P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}] }{P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}]}\] 


less than some quite small \(\gamma\). This condition is only needed
because our measure of evidential distinguishability, QI, blows up
when the ratio 

 

\[\frac{P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}]}{P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}]}\]


is extremely small. Furthermore, this condition is really no
restriction at all on possible experiments or observations. If \(c\_k\)
has some possible outcome sentence \(o\_{ku}\) that would make 



\[\frac{P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}]}{P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}]} \lt \gamma\] 


(for a given small \(\gamma\) of interest), one may disjunctively lump
\(o\_{ku}\) together with some other outcome sentence \(o\_{kv}\) for
\(c\_k\). Then, the antecedent condition of the theorem will be
satisfied, but with the sentence '\((o\_{ku} \vee
o\_{kv})\)' treated as a single outcome. It can be proved that
the only effect of such "disjunctive lumping" is to make
\(\bEQI\) smaller than it would otherwise be (whereas larger values of
\(\bEQI\) are more desirable). If the *too strongly refuting*
disjunct \(o\_{ku}\) actually occurs when the experiment or observation
\(c\_k\) is conducted, all the better, since this results in a
likelihood ratio 

 

\[\frac{P[o\_{ku} \pmid h\_{j}\cdot b\cdot c\_{k}]}{P[o\_{ku} \pmid h\_{i}\cdot b\cdot c\_{k}]}\]


smaller than \(\gamma\) on that particular evidential outcome. We
merely failed to take this *more strongly refuting* possibility
into account when computing our lower bound on the *likelihood that
refutation via likelihood ratios* would occur.



The point of the two *Convergence Theorems* explored in this
section is to assure us, in advance of the consideration of any
specific pair of hypotheses, that if the possible evidence streams
that test them have certain characteristics which reflect their
evidential distinguishability, it is highly likely that outcomes
yielding small likelihood ratios will result. These theorems provide
finite lower bounds on how quickly convergence is likely to occur.
Thus, there is no need to wait through some infinitely long run for
convergence to occur. Indeed, for any evidence sequence on which the
probability distributions are at all well behaved, the *actual
likelihood* of obtaining outcomes that yield small likelihood
ratio values will inevitably be *much higher* than the lower
bounds given by Theorems 1 and 2.



In sum, according to Theorems 1 and 2, each hypothesis \(h\_i\)
*says*, via likelihoods, that given enough observations,
*it* is very likely to dominate its empirically distinct rivals
in a contest of likelihood ratios. The true hypothesis speaks
truthfully about this, and its competitors lie. Even a sequence of
observations with an extremely low *average expected quality of
information* is very likely to do the job if that evidential
sequence is long enough. Thus (by
 Equation 9\*),
 as evidence accumulates, the *degree of support* for false
hypotheses will very probably approach 0, indicating that they are
probably false; and as this happens, (by Equations 10 and 11) the
degree of support for the true hypothesis will approach 1, indicating
its probable truth. Thus, the **Criterion of Adequacy**
(CoA) is satisfied.


## 5. When the Likelihoods are Vague or Diverse



Up to this point we have been supposing that likelihoods possess
objective or agreed numerical values. Although this supposition is
often satisfied in scientific contexts, there are important settings
where it is unrealistic, where hypotheses only support vague
likelihood values, and where there is enough ambiguity in what
hypotheses *say* about evidential claims that the scientific
community cannot agree on precise values for the likelihoods of
evidential
 claims.[18]
 Let us now see how the supposition of precise, agreed likelihood
values may be relaxed in a reasonable way.



Recall why agreement, or near agreement, on precise values for
likelihoods is so important to the scientific enterprise. To the
extent that members of a scientific community disagree on the
likelihoods, they disagree about the empirical content of their
hypotheses, about what each hypothesis *says* about how the
world is likely to be. This can lead to disagreement about which
hypotheses are refuted or supported by a given body of evidence.
Similarly, to the extent that the values of likelihoods are only
vaguely implied by hypotheses as understood by an individual agent,
that agent may be unable to determine which of several hypotheses is
refuted or supported by a given body of evidence.



We have seen, however, that the individual values of likelihoods are
not really crucial to the way evidence impacts hypotheses. Rather, as
Equations 9-11 show, it is *ratios of likelihoods* that
do the heavy lifting. So, even if two support functions \(P\_{\alpha}\)
and \(P\_{\beta}\) disagree on the values of individual likelihoods,
they may, nevertheless, largely agree on the refutation or support
that accrues to various rival hypotheses, provided that the following
condition is satisfied: 



**Directional Agreement Condition**:
  

The likelihood ratios due to each of a pair of support functions
\(P\_{\alpha}\) and \(P\_{\beta}\) are said to *agree in
direction* (with respect to the possible outcomes of experiments
or observations relevant to a pair of hypotheses) *just in
case*
* whenever possible outcome sequence \(e^n\) makes 

\[\frac{P\_{\alpha}[e^n \pmid h\_{j}\cdot b\cdot c^{n}]}{P\_{\alpha}[e^n \pmid h\_{i}\cdot b\cdot c^{n}]} \lt 1,\]

 it
also makes 

\[\frac{P\_{\beta}[e^n \pmid h\_{j}\cdot b\cdot c^{n}]}{P\_{\beta}[e^n \pmid h\_{i}\cdot b\cdot c^{n}]} \lt 1;\]
* whenever possible outcome sequence \(e^n\) makes 

\[\frac{P\_{\alpha}[e^n \pmid h\_{j}\cdot b\cdot c^{n}]}{P\_{\alpha}[e^n \pmid h\_{i}\cdot b\cdot c^{n}]} \gt 1,\]

 it
also makes 

\[\frac{P\_{\beta}[e^n \pmid h\_{j}\cdot b\cdot c^{n}]}{P\_{\beta}[e^n \pmid h\_{i}\cdot b\cdot c^{n}]} \gt 1;\]
* each of these likelihood ratios is either close to 1 for both of
these support functions, or is quite far from 1 for both of
 them.[19]




When this condition holds, the evidence will support \(h\_i\) over
\(h\_j\) according to \(P\_{\alpha}\) just in case it does so for
\(P\_{\beta}\) as well, although the strength of support may differ.
Furthermore, although the *rate* at which the likelihood ratios
increase or decrease on a stream of evidence may differ for the two
support functions, the impact of the cumulative evidence should
ultimately affect their refutation or support in much the same way.




When likelihoods are vague or diverse, we may take an approach similar
to that we employed for *vague* and *diverse* prior
plausibility assessments. We may extend the *vagueness sets*
for individual agents to include a collection of inductive support
functions that cover the range of values for likelihood ratios of
evidence claims (as well as cover the ranges of comparative support
strengths for hypotheses due to plausibility arguments within
*b*, as represented by ratios of prior probabilities). Similarly,
we may extend the *diversity sets* for communities of agents to
include support functions that cover the ranges of likelihood ratio
values that arise within the *vagueness sets* of members of the
scientific community.



This broadening of *vagueness* and *diversity* sets to
accommodate vague and diverse likelihood values makes no trouble for
the *convergence to truth results* for hypotheses. For,
provided that the *Directional Agreement Condition* is
satisfied by all support functions in an extended *vagueness*
or *diversity set* under consideration, the *Likelihood
Ratio Convergence Theorem* applies to each individual support
function in that set. For, the the proof of that convergence theorem
doesn't depend on the supposition that likelihoods are objective
or have intersubjectively agreed values. Rather, it applies to each
individual support function \(P\_{\alpha}\). The only possible problem
with applying this result across a range of support functions is that
when their values for likelihoods differ, function \(P\_{\alpha}\) may
disagree with \(P\_{\beta}\) on which of the hypotheses is favored by a
given sequence of evidence. That can happen because different support
functions may represent the evidential import of hypotheses
differently, by specifying different likelihood values for the very
same evidence claims. So, an evidence stream that favors \(h\_i\)
according to \(P\_{\alpha}\) may instead favor \(h\_j\) according to
\(P\_{\beta}\). However, when the *Directional Agreement
Condition* holds for a given collection of support functions, this
problem cannot arise. *Directional Agreement* means that the
evidential import of hypotheses is similar enough for \(P\_{\alpha}\)
and \(P\_{\beta}\) that a sequence of outcomes may favor a hypothesis
according to \(P\_{\alpha}\) only if it does so for \(P\_{\beta}\) as
well.



Thus, when the *Directional Agreement Condition* holds for all
support functions in a *vagueness* or *diversity* set
that is extended to include vague or diverse likelihoods, and provided
that enough evidentially distinguishing experiments or observations
can be performed, all support functions in the extended
*vagueness* or *diversity* set will very probably come
to agree that the likelihood ratios for empirically distinct false
competitors of a true hypothesis are extremely small. As that happens,
the community comes to agree on the refutation of these competitors,
and the true hypothesis rises to the top of the
 heap.[20]




What if the true hypothesis has evidentially equivalent rivals? Their
posterior probabilities must rise as well. In that case we are only
assured that the disjunction of the true hypothesis with its
evidentially equivalent rivals will be driven to 1 as evidence lays
low its evidentially distinct rivals. The true hypothesis will itself
approach 1 only if either it has no evidentially equivalent rivals, or
whatever equivalent rivals it does have can be laid low by
plausibility arguments of a kind that don't depend on the
evidential likelihoods, but only show up via the comparative
plausibility assessments represented by ratios of prior
probabilities.