## 1. Motivation and Preliminaries



In this section, we will provide some motivation for trying to
understand causation in terms of probabilities, and address a couple
of preliminary issues.


### 1.1 Problems for Regularity Theories



According to David Hume, causes are invariably followed by their
effects: 



> 
> 
> We may define a cause to be *an object, followed by another, and
> where all the objects similar to the first, are followed by objects
> similar to the second*. (1748: section VII) 
> 
> 
> 



Attempts to analyze causation in terms of invariable patterns of
succession are referred to as "regularity theories" of
causation. There are a number of well-known problems facing regularity
theories, at least in their simplest forms, and these may be used to
motivate probabilistic approaches to causation. Moreover, an overview
of these difficulties will help to give a sense of the kinds of
problem that any adequate theory of causation would have to solve.



(i) *Imperfect Regularities*. The first difficulty is that most
causes are not invariably followed by their effects. For example,
smoking is a cause of lung cancer, even though some smokers do not
develop lung cancer. Imperfect regularities may arise for two
different reasons. First, they may arise because of the
*heterogeneity* of circumstances in which the cause arises. For
example, some smokers may have a genetic susceptibility to lung
cancer, while others do not; some non-smokers may be exposed to other
carcinogens (such as asbestos), while others are not. Second,
imperfect regularities may also arise because of a failure of
*physical determinism*. If an event is not determined to occur,
then no other event can be (or be a part of) a sufficient condition
for that event. The success of quantum mechanics--and to a lesser
extent, other theories employing probability--has shaken our
faith in determinism. Thus it has struck many philosophers as
desirable to develop a theory of causation that does not presuppose
determinism.



The central idea behind probabilistic theories of causation is that
causes *change the probability* of their effects; an effect may
still occur in the absence of a cause or fail to occur in its
presence. Thus smoking is a cause of lung cancer, not because all
smokers develop lung cancer, but because smokers are *more
likely* to develop lung cancer than non-smokers. This is entirely
consistent with there being some smokers who avoid lung cancer, and
some non-smokers who succumb to it.



(ii) *Irrelevance*. A condition that is invariably followed by
some outcome may nonetheless be irrelevant to that outcome. Salt that
has been hexed by a sorcerer invariably dissolves when placed in water
(Kyburg 1965), but hexing does not cause the salt to dissolve. Hexing
does not *make a difference for* dissolution. Probabilistic
theories of causation capture this notion of making a difference by
requiring that a cause make a difference for the probability of its
effect.



(iii) *Asymmetry*. If *A* causes *B*, then,
typically, *B* will not also cause *A*. Smoking causes lung
cancer, but lung cancer does not cause one to smoke. One way of
enforcing the asymmetry of causation is to stipulate that causes
precede their effects in time. But it would be nice if a theory of
causation could provide some explanation of the directionality of
causation, rather than merely stipulate it. Some proponents of
probabilistic theories of causation have attempted to use the
resources of probability theory to articulate a substantive account of
the asymmetry of causation.



(iv) *Spurious Regularities*. Suppose that a cause is regularly
followed by two effects. Here is an example from Jeffrey (1969):
Suppose that whenever the barometric pressure in a certain region
drops below a certain level, two things happen. First, the height of
the column of mercury in a particular barometer drops below a certain
level. Shortly afterwards, a storm occurs. This situation is shown
schematically in Figure 1. Then, it may well also be the case that
whenever the column of mercury drops, there will be a storm. If so, a
simple regularity theory would seem to rule that the drop of the
mercury column *causes* the storm. In fact, however, the
regularity relating these two events is *spurious*. The ability
to handle such spurious correlations is probably the greatest source
of attraction for probabilistic theories of causation.



![A diagram with three text nodes. The base one is 'Drop in Barometric Pressure' with arrows pointing to 'Drop in Column of Mercury' and 'Storm'.](figure1.svg)

Figure 1



### 1.2 Probability



In this sub-section, we will review some of the basics of the
mathematical theory of probability, and introduce some notation.
Readers already familiar the mathematics of probability may wish to
skip this section.



Probability is a function, P, that assigns values between zero and
one, inclusive. Usually the arguments of the function are taken to be
sets, or propositions in a formal language. The formal term for these
arguments is 'events'. We will here use the notation that
is appropriate for propositions, with '\(\nsim\)'
representing negation, '&' representing conjunction,
and '\(\vee\)' representing disjunction. Sometimes when
there is a long conjunction, this is abbreviated by using commas instead
of ampersands. The domain of a probability function has the structure
of a *field* or a *Boolean algebra*. This means that the
domain is closed under complementation and the taking of finite unions
or intersections (for sets), or under negation, conjunction, and
disjunction (for propositions). Thus if *A* and *B* are
events in the domain of P, so are \({\nsim}A\), \(A \amp B\), and \(A
\vee B\).



Some standard properties of probability are the following:


* If *A* is a contradiction, then \(\PP(A) = 0\).
* If *A* is a tautology, then \(\PP(A) = 1\).
* If \(\PP(A \amp B) = 0\), then \(\PP(A \vee B) = \PP(A) +
\PP(B)\)
* \(\PP({\nsim}A) = 1 - \PP(A)\).



In addition to probability theory, the entry will use basic notation from set
theory and logic. Sets will appear in boldface.


* \(\varnothing\) is the empty set (the set that has no members or
elements)
* \(x \in \bX\) means that *x* is a member or element of the
set \(\bX\).
* \(\bX \subseteq \bY\) means that \(\bX\) is a subset of \(\bY\);
i.e., every member of \(\bX\) is also a member of \(\bY\). Note that
both \(\varnothing\) and \(\bY\) are subsets of \(\bY\).
* \(\bX \setminus \bY\) is the set that results from removing the
members of \(\bY\) from \(\bX\).
* \(\forall\) and \(\exists\) are the universal and existential
quantifiers, respectively.



Some further definitions:


* The *conditional probability of A given B,* written \(\PP(A
\mid B)\), is standardly defined as follows: 

\[
\PP(A \mid B) = \frac{\PP(A \amp B)}{\PP(B)}.
\]


If \(\PP(B) = 0\), then the ratio in the definition of conditional
probability is undefined. There are, however, a variety of technical
developments that will allow us to define \(\PP(A \mid B)\) when
\(\PP(B)\) is 0. We will ignore this problem here.
* A *random variable* for probability P is a function
*X* that takes values in the real numbers, such that for any
number \(x, X = x\) is an event in the domain of P. For example, we
might have a random variable \(T\_1\) that takes values in \(\{1, 2, 3,
4, 5, 6\}\), representing the outcome of the first toss of a die. The
event \(T\_1 = 3\) would represent the first toss as having outcome 3.
We will write \(T\_1 \in \{1, 2, 3\}\) as shorthand for \(T\_1 = 1 \vee
T\_1 = 2 \vee T\_1 = 3\).
* If *A* and *B* are in the domain of P, then *A* and
*B* are *probabilistically independent* (with respect to
P) just in case \(\PP(A \amp B) = \PP(A) \times \PP(B). A\) and
*B* are *probabilistically dependent* or
*correlated* otherwise.
* *A* and *B* are *probabilistically independent
conditional on* *C* if and only if \(\PP(A \amp B
\mid C) = \PP(A \mid C) \times \PP(B \mid C)\).
* Random variables *X* and *Y* are *probabilistically
independent* if and only if all events of the form \(X \in \bH\)
are probabilistically independent of all events of the form \(Y \in
\bJ\), where \(\bH\) and \(\bJ\) are subsets of the range of *X*
and *Y*, respectively.



As a convenient shorthand, a probabilistic statement that contains
only a variable or set of variables, but no values, will be understood
as a universal quantification over all possible values of the
variable(s). Thus if \(\bX = \{X\_1 , \ldots ,X\_m\}\) and \(\bY = \{Y\_1
, \ldots ,Y\_n\}\), we may write

 

\[
\PP(\bX \mid \bY) = \PP(\bX)
\]


as shorthand for

 

\[\begin{align} 
\forall x\_1 \ldots \forall x\_m\forall y\_1 \ldots \forall y\_n & [\PP(X\_1 =x\_1 ,\ldots ,X\_m =x\_m \mid Y\_1 =y\_1,\ldots ,Y\_n =y\_n)\\ 
& = \PP(X\_1 =x\_1 ,\ldots ,X\_m =x\_m)]
\end{align}\]


(where the domain of quantification for each variable will be the
range of the relevant random variable).


### 1.3 The Interpretation of Probability



Causal relations are normally thought to be objective features of the
world. If they are to be captured in terms of probability theory, then
probability assignments should represent some objective feature of the
world. There are a number of attempts to interpret probabilities
objectively, the most prominent being *frequency*
interpretations and *propensity* interpretations. Most
proponents of probabilistic theories of causation have understood
probabilities in one of these two ways. Notable exceptions are Suppes
(1970), who takes probability to be a feature of a model of a
scientific theory; and Skyrms (1980), who understands the relevant
probabilities to be the subjective probabilities of a certain kind of
rational agent.


### 1.4 General Causation and Actual Causation



It is common to distinguish between *general*, or
*type-level* causation, on the one hand, and *singular*,
*token-level* or *actual* causation, on the other. This
entry adopts the terms *general causation* and *actual
causation.* Causal claims usually have the structure
'*C* causes *E*'. *C* and *E* are the
*relata* of the causal claim; we will discuss causal relata in
more detail in the next section. General causation and actual
causation are often distinguished by their relata. *General*
causal claims, such "smoking causes lung cancer" typically
do not refer to particular individuals, places, or times, but only to
event-types or properties. *Singular* causal claims, such as
"Jill's heavy smoking during the 2000s caused her to
develop lung cancer", typically do make reference to particular
individuals, places, and times. This is an imperfect guide, however;
for example, some theories of general causation to be discussed below
take their causal relata to be time-indexed.



A related distinction is that general causation is concerned with a
full range of possibilities, whereas actual causation is concerned
with how events actually play out in a specific case. At a minimum, in
claims of actual causation, "cause" functions as a success
verb. The claim "Jill's heavy smoking during the 2000s
caused her to develop lung cancer" implies that Jill smoked
heavily during the 2000s and that she developed lung cancer.



The theories to be discussed in Sections
 2
 and
 3
 below primarily concern general causation, while
 Section 4
 discusses theories of actual causation.


### 1.5 Causal Relata



A number of different candidates have been proposed for the relata of
causal relations. The relata of actual causal relations are often
taken to be *events* (not to be confused with events in the
purely technical sense), although some authors (e.g., Mellor 2004)
argue that they are *facts*. The relata of general causal
relations are often taken to be *properties* or
*event-types.* For purposes of definiteness, 
*events* will refer to the relata of actual causation, and
*factors* will refer to the relata of general causation. These
terms are not intended to imply a commitment to any particular view on
the nature of the causal relata.



In probabilistic approaches to causation, causal relata are
represented by events or random variables in a probability space.
Since the formalism requires us to make use of negation, conjunction,
and disjunction, the relata must be entities (or be accurately
represented by entities) to which these operations can be meaningfully
applied.



In some theories, the time at which an event occurs or a property is
instantiated plays an important role. In such cases, it will be useful
to include a subscript indicating the relevant time. Thus the relata
might be represented by \(C\_t\) and \(E\_{t'}\). If the relata are
particular events, this subscript is just a reminder; it adds no
further information. For example, if the event in question is the
opening ceremony of the Rio Olympic games, the subscript
'8/5/2016' is not necessary to disambiguate it from other
events. In the case of properties or event-types, however, such
subscripts do add further information. The time index need not refer
to a date or absolute time. It could refer to a stage in the
development of a particular kind of system. For example, exposure to
lead paint in children can cause learning disabilities. Here the time
index would indicate that it is exposure *in children*, that
is, in the early stages of human life, that causes the effect in
question. The time indices may also indicate relative times. Exposure
to the measles virus causes the appearance of a rash approximately two
weeks later. We might indicate this time delay by assigning exposure a
time index of \(t = 0\), and rash an index of \(t = 14\) (for 14
days).



It is standard to assume that causes and effects must be
*distinct* from one another. This means that they must not
stand in logical relations or part-whole relations to one another.
Lewis 1986a contains a detailed discussion of the relevant notion of
distinctness. We will typically leave this restriction tacit.


### 1.6 Further Reading



Psillos 2009 provides an overview of regularity theories of causation.
Lewis 1973 contains a brief but clear and forceful overview of
problems with regularity theories. The entry for
 scientific explanation
 contains discussions of some of these problems.



Hajek and Hitchcock 2016b is a short introduction to probability
theory geared toward philosophical applications. Billingsley 1995 and
Feller 1968 are two standard texts on probability theory. The entry
for
 interpretations of probability
 includes a brief introduction to the formalism of probability theory,
and discusses the various interpretations of probability. Galavotti
2005 and Gillies 2000 are good surveys of philosophical theories of
probability. Hajek and Hitchcock 2016a includes essays covering the
major interpretations of probability. 



The Introduction of Eells 1991 provides a good overview of the
distinction between general and actual causation.



Bennett 1988 is an excellent discussion of facts and events in the
context of causation. Ehring 2009 is a survey of views about causal
relata. See also the entries for
 the metaphysics of causation,
 events,
 facts,
 and
 properties.


## 2. Probability-raising Theories of Causation



The theories canvassed in this section all develop the basic idea that
causes raise the probability of their effects. These theories were
among the leading theories of causation during the second half of the
20th century. Today, they have largely been supplanted by
the causal modeling approaches discussed in
 Section 3.


### 2.1 Probability-raising and Conditional Probability



The central idea that causes raise the probability of their effects
can be expressed formally using conditional probability. *C*
raises the probability of *E* just in case:



\((\PR\_1)\)
\(\PP(E \mid C) \gt \PP(E)\).


In words, the probability that *E* occurs, given that *C*
occurs, is higher than the unconditional probability that *E*
occurs. Alternately, we might say that *C* raises the probability
of *E* just in case:



\((\PR\_2)\)
\( \PP(E \mid C) \gt \PP(E \mid {\nsim}C)\);


the probability that *E* occurs, given that *C* occurs, is
higher than the probability that *E* occurs, given that *C*
does not occur. These two formulations turn out to be equivalent in
the sense that inequality \(\PR\_1\) will hold just in case \(\PR\_2\)
holds. Some authors (e.g., Reichenbach 1956, Suppes 1970, Cartwright
1979) have formulated probabilistic theories of causation using
inequalities like \(\PR\_1\), others (e.g., Skyrms 1980, Eells 1991)
have used inequalities like \(\PR\_2\). This difference is mostly
immaterial, but for consistency we will stick with (\(\PR\_2)\). Thus a
first stab at a probabilistic theory of causation would be:



(PR)
*C* is a
cause of *E* just in case \(\PP(E \mid C) \gt \PP(E \mid
{\nsim}C)\).


PR has some advantages over the simplest version of a regularity
theory of causation (discussed in
 Section 1.1
 above). PR is compatible with imperfect regularities: *C* may
raise the probability of *E* even though instances of *C*
are not invariably followed by instances of *E*. Moreover, PR
addresses the problem of relevance: if *C* is a cause of
*E*, then *C* makes a difference for the probability of
*E*. But as it stands, PR does not address either the problem of
asymmetry, or the problem of spurious correlations. PR does not
address the problem of asymmetry because probability-raising turns out
to be symmetric: \(\PP(E \mid C) \gt \PP(E \mid {\nsim}C)\),
if and only if \(\PP(C \mid E) \gt \PP(C \mid {\nsim}E)\).
Thus PR by itself cannot determine whether *C* is the cause of
*E* or vice versa. PR also has trouble with *spurious
correlations*. If *C* and *E* are both caused by some
third factor, *A*, then it may be that \(\PP(E \mid C) \gt
\PP(E \mid {\nsim}C)\) even though *C* does not cause
*E*. This is the situation shown in
 Figure 1
 above. Here, *C* is the drop in the level of mercury in a
barometer, and *E* is the occurrence of a storm. Then we would
expect that \(\PP(E \mid C) \gt \PP(E \mid {\nsim}C)\). In
this case, atmospheric pressure is referred to as a
*confounding* factor.


### 2.2 Screening off



Hans Reichenbach's *The Direction of Time* was published
posthumously in 1956. In it, Reichenbach is concerned with the origins
of temporally asymmetric phenomena, particularly the increase in
entropy dictated by the second law of thermodynamics. In this work, he
presents the first fully developed probabilistic theory of causation,
although some of the ideas can be traced back to an earlier paper from
1925 (Reichenbach 1925).



Reichenbach introduced the terminology of *screening off* to
describe a particular type of probabilistic relationship. If \(\PP(E
\mid A \amp C) = \PP(E \mid C)\), then *C* is said to
screen *A* off from *E*. When \(\PP(A \amp C) \gt 0\), this
equality is equivalent to \(\PP(A \amp E \mid C) = \PP(A \mid
C) \times \PP(E \mid C)\); i.e., *A* and *E* are
probabilistically independent conditional upon *C*.



Reichenbach recognized that there were two kinds of causal structure
in which *C* will typically screen *A* off from *E*.
The first occurs when *A* causes *C*, which in turn causes
*E*, and there is no other route or process by which *A*
effects *E*. This is shown in Figure 2.



![The letter 'A' with an arrow to the right pointing to the letter 'C' which in turn has an arrow to the right pointing to the letter 'E'](figure2.svg)

Figure 2




In this case, Reichenbach said that *C* is *causally
between* *A* and *E*. We might say that *C* is an
intermediate cause between *A* and *E*, or that *C* is
a proximate cause of *E* and *A* a distal cause of *E*.
For example, unprotected sex (*A*) causes AIDS (*E*) only by
causing HIV infection (*C*). Then we would expect that among
those already infected with HIV, those who became infected through
unprotected sex would be no more likely to contract AIDS than those
who became infected in some other way.



The second type of case that produces screening off occurs when
*C* is a common cause of *A* and *E*, such as in the
barometer example depicted in
 Figure 1.
 A drop in atmospheric pressure (*C*) causes both a drop in the
level of mercury in a barometer (*A*) and a storm (*E*).
(This notation is slightly different from one used earlier.) The
atmospheric pressure will screen off the barometer reading from the
weather: given that the atmospheric pressure has dropped, the
reading of the barometer makes no difference for the probability of
whether a storm will occur.



Reichenbach used the apparatus of screening off to address the problem
of spurious correlations. In our example, while a drop in the column
of mercury (*A*) raises the probability of a storm (*E*)
overall, it does not raise the probability of a storm when we further
condition on the atmospheric pressure. That is, if *A* and
*E* are spuriously correlated, then *A* will be screened off
from *E* by a common cause. More specifically, suppose that
\(C\_t\) and \(E\_{t'}\) are events that occur at times *t* and
\(t'\) respectively. Then



(Reich)
\(C\_t\) is a cause of \(E\_{t'}\) if and only if:
 
i.
\(t \lt t'\)
ii.
\(\PP(E\_{t'} \mid C\_t) \gt \PP(E\_{t'} \mid {\nsim}C\_t)\); and
iii.
There is no further event *B\(\_{t''}\)*, occurring at 
 a time \(t''\) earlier than or simultaneously with *t*, 
 that screens \(E\_{t'}\) off from \(C\_t\).




Note the restriction of \(t''\) to times earlier than or
simultaneously with the occurrence of \(C\_t\). That is because causal
intermediates between \(C\_t\) and \(E\_{t'}\) will often screen \(C\_t\)
off from \(E\_{t'}\). In such cases we still want to say that \(C\_t\)
is a cause of \(E\_{t'}\), albeit a distal or indirect cause.



Suppes (1970) independently offered an equivalent definition of
causation, although his motivation for the no-screening-off condition
was different from Reichenbach's. Suppes extended the framework
in a number of directions. While Reichenbach was interested in
probabilistic causation primarily in connection with issues that arise
within the foundations of statistical mechanics, Suppes was interested
in defining causation within the framework of probabilistic models of
scientific theories. For example, Suppes offers an extended discussion
of causation in the context of psychological models of learning.


### 2.3 The Common Cause Principle



Reichenbach (1956) formulated a principle he dubbed the 'Common Cause
Principle' (CCP). Suppose that events *A* and *B* are
positively correlated, i.e., that


1. \(\PP(A \amp B) \gt \PP(A) \times \PP(B)\).



But suppose that neither *A* nor *B* is a cause of the
other. Then Reichenbach maintained that there will be a common cause,
*C*, of *A* and *B*, satisfying the following
conditions:


2. \(0 \lt \PP(C) \lt 1\)
3. \(\PP(A \amp B \mid C) = \PP(A \mid C) \times \PP(B
\mid C)\)
4. \(\PP(A \amp B \mid {\nsim}C) = \PP(A \mid {\nsim}C)
\times \PP(B \mid {\nsim}C)\)
5. \(\PP(A \mid C) \gt \PP(A \mid {\nsim}C)\)
6. \(\PP(B \mid C) \gt \PP(B \mid {\nsim}C)\).



When events *A*, *B*, and *C* satisfy these conditions,
they are said to form a *conjunctive fork*. 5 and 6 follow from
*C* being a cause of *A* and a cause of *B*. Conditions
2 and 3 stipulate that *C* and \({\nsim}C\) screen off *A*
from *B*.



Conditions 2 through 6 mathematically entail 1. Reichenbach says that
the common cause *explains* the correlation between *A*
and *B*. The idea is that probabilistic correlations that are not
the result of one event causing another are ultimately derived from
probabilistic correlations that do result from a causal
relationship.



Reichenbach's definition of causation, discussed in
 Section 2.2
 above, appeals to time order: it requires that a cause occur earlier
than its effect. But Reichenbach also thought that the direction from
causes to effects can be identified with a pervasive statistical
asymmetry. Suppose that events *A* and *B* are correlated,
and that *C* satisfies conditions 2-6 above, so that
*ABC* form a conjunctive fork. If *C* occurs earlier than
*A* and *B*, and there is no event satisfying 2 through 6
that occurs later than *A* and *B*, then *ACB* is said
to form a conjunctive fork *open to the future*. Analogously,
if there is a later event satisfying 2 through 6, but no earlier
event, we have a conjunctive fork *open to the past*. If an
earlier event *C* and a later event *D* both satisfy 2
through 6, then *ACBD* forms a *closed fork*.
Reichenbach's proposal was that the direction from cause to
effect is the direction in which open forks predominate. In our world,
there are a great many forks open to the future, few or none open to
the past. However, we shall see in
 section 3.6
 below that conjunctive forks are not the best structures for
identifying causal direction. 


### 2.4 Simpson's Paradox and Background Contexts



In the Reichenbach-Suppes definition of causation, the inequality
\(\PP(E\_{t'} \mid C\_t) \gt \PP(E\_{t'} \mid {\nsim}C\_t)\) is
necessary, but not sufficient, for causation. It is not sufficient,
because it may hold in cases where \(C\_t\) and \(E\_{t'}\) share a
common cause. Unfortunately, common causes can also give rise to cases
where this inequality is not necessary for causation either. Suppose,
for example, that smoking is highly correlated with living in the
country: those who live in the country are much more likely to smoke
as well. Smoking is a cause of lung cancer, but suppose that city
pollution is an even stronger cause of lung cancer. Then it may be
that smokers are, over all, less likely to suffer from lung cancer
than non-smokers. Letting *C* represent smoking, *B* living
in the country, and *E* lung cancer, \(\PP(E \mid C) \lt
\PP(E \mid {\nsim}C)\). Note, however, that if we conditionalize
on whether one lives in the country or in the city, this inequality is
reversed: \(\PP(E \mid C \amp B) \gt \PP(E \mid {\nsim}C \amp
B)\), and \(\PP(E \mid C \amp{\nsim}B) \gt \PP(E \mid {\nsim}C
\amp{\nsim}B)\). Such reversals of probabilistic inequalities are
instances of "Simpson's Paradox". The problem that
Simpson's paradox creates for probabilistic theories of
causation was pointed out by Nancy Cartwright (1979) and Brian Skyrms
(1980) at about the same time.



Cartwright and Skyrms sought to rectify the problem by replacing
conditions (ii) and (iii) of
 Reich
 with the requirement that causes must raise the probabilities of
their effects in various *background contexts*. Cartwright
proposed the following definition:



(Cart)
*C* causes *E* if and only if 
 \(\PP(E \mid C \amp B) \gt \PP(E \mid {\nsim}C \amp B)\) 
 for every background context *B*.


Skyrms proposed a slightly weaker condition: a cause must raise the
probability of its effect in at least one background context, and
lower it in none. A background context is a conjunction of factors.
When such a conjunction of factors is conditioned on, those factors
are said to be "held fixed". To specify what the
background contexts will be, then, we must specify what factors are to
be held fixed. In the previous example, we saw that the true causal
relevance of smoking for lung cancer was revealed when we held country
living fixed, either positively (conditioning on \(B)\) or negatively
(conditioning on \({\nsim}B)\). This suggests that in evaluating the
causal relevance of *C* for *E*, we need to hold fixed other
causes of *E*, either positively or negatively. This suggestion
is not entirely correct, however. Let *C* and *E* be smoking
and lung cancer, respectively. Suppose *D* is a causal
intermediary, say the presence of tar in the lungs. If *C* causes
*E* exclusively via *D*, then *D* will screen *C*
off from *E*: given the presence (absence) of tar in the lungs,
the probability of lung cancer is not affected by whether the tar got
there by smoking. Thus we will not want to hold fixed any causes of
*E* *that are themselves caused by C*. Let us call the set
of all factors that are causes of *E*, but are not caused by
*C*, the set of *independent* causes of *E*. A
background context for *C* and *E* will then be a maximal
conjunction, each of whose conjuncts is either an independent cause of
*E*, or the negation of an independent cause of *E*.



Note that the specification of factors that need to be held fixed
appeals to causal relations, so the theory no longer offers a
*reductive analysis* of causation. Nonetheless, the theory
imposes probabilistic constraints upon possible causal relations in
the sense that a given set of probability relations will be
incompatible with at least some systems of causal relations. Note also
that we have dropped the subscripts referring to times. Cartwright
claimed that it is not necessary to appeal to the time order of events
to distinguish causes from effects in her theory. That is because it
will no longer be true in general that if *C* raises the
probability of *E* in every relevant background context *B*,
then *E* raise will raise the probability of *C* in every
background context \(B'\). The reason is that the construction of the
background contexts ensures that the background contexts relevant to
assessing *C*'s causal relevance for *E* are different
from those relevant to assessing *E*'s causal relevance for
*C*. However, Davis (1988) and Eells (1991) both argue cogently
that Cartwright's account will still sometimes rule that effects
bring about their causes.


### 2.5 Other Causal Relations



Cartwright defined a cause as a factor that increases the probability
of its effect in every background context. But it is easy to see that
there are other possible probability relations between *C* and
*E*. Eells (1991) proposes the following taxonomy:



(Eells)


*
\(C\_t\) is a
 *positive cause* (or simply *cause*) of \(E\_{t'}\) if
 and only if \(t \lt t'\) and \(\PP(E\_{t'} \mid C\_t \amp B) \gt
 \PP(E\_{t'} \mid {\nsim}C\_t \amp B)\) for every background context
 *B*.
*
\(C\_t\) is a
 *negative cause* of \(E\_{t'}\) (or \(C\_t\) *prevents*
 \(E\_{t'}\) or \(C\_t\) *inhibits* \(E\_{t'})\) if and only if \(t
 \lt t'\) and \(\PP(E\_{t'} \mid C\_t \amp B) \lt \PP(E\_{t'} \mid
 {\nsim}C\_t \amp B)\) for every background context *B*.
*
\(C\_t\) is
 *causally neutral* for \(E\_{t'}\) (or *causally
 irrelevant* for \(E\_{t'})\) if and only if \(\PP(E\_{t'} \mid
 C\_t \amp B) = \PP(E\_{t'} \mid {\nsim} C\_t \amp B)\) for every
 background context *B*.
*
\(C\_t\) is a
 *mixed cause* (or *interacting cause*) of \(E\_{t'}\) if
 and only if \(t \lt t'\) and \(C\_t\) *is not a positive or negative
 cause of* \(E\_{t'}\) and \(C\_t\) *is not causally neutral
 for* \(E\_{t'}\).

\(C\_t\) is *causally relevant* for \(E\_{t'}\) if and only if it
is a positive, negative, or mixed cause of \(E\_{t'}\); i.e., if and
only if \(t \lt t'\) and \(C\_t\) is not causally neutral for
\(E\_{t'}\).



It should be apparent that when constructing background contexts for
*C* and *E* one should hold fixed not only (positive) causes
of *E* that are independent of \(C,\) but also negative and mixed
causes of *E*; in other words, one should hold fixed all factors
that are causally relevant for *E*, except those for which
*C* is causally relevant. This suggests that causal relevance,
rather than positive causation, is the most basic metaphysical
concept.



Eells's taxonomy brings out an important distinction. It is one
thing to ask whether *C* is causally relevant to *E* *in
some way*; it is another to ask *in which way* *C* is
causally relevant to *E*. To say that *C* causes *E* is
then potentially ambiguous: it might mean that *C* is causally
relevant to *E*; or it might mean that *C* is a positive
cause of *E*. Probabilistic theories of causation can be used to
answer both types of question.



Eells claims that general causal claims must be relativized to a
population. A very heterogeneous population will include a great many
different background conditions, while a homogeneous population will
contain few. A heterogeneous population can always be subdivided into
homogeneous subpopulations. It will often happen that *C* is a
mixed cause of *E* relative to a population P, while being a
positive cause, negative cause, or causally neutral for *E* in
various subpopulations of P.


### 2.6 Contextual-unanimity



According to both
 Cart
 and
 Eells,
 a cause must raise the probability of its effect in *every*
background context. This has been called the requirement of
*contextual-unanimity*. Dupre (1984) raises the following
counterexample to the contextual unanimity requirement. Suppose that
there is a very rare gene that has the following effect: those that
possess the gene have their chances of contracting lung cancer
*lowered* when they smoke. In this scenario, there would be a
background context in which smoking lowers the probability of lung
cancer: thus smoking would not be a cause of lung cancer according to
the contextual-unanimity requirement. Nonetheless, it seems unlikely
that the discovery of such a gene would lead us to abandon the claim
that smoking causes lung cancer. 



Dupre suggests instead that we should deem *C* to be a cause of
*E* if it raises the probability of *E* in a 'fair
sample'--a sample that is representative of the population
as a whole. Mathematically, this amounts to the requirement that




(Dupre)
*C* causes
*E* if and only if

 \[\Sigma\_B \PP(E \mid C \amp B) \times \PP(B) \gt \Sigma\_B \PP(E \mid {\nsim}C \amp B) \times \PP(B)\]



where *B* ranges over the relevant background contexts. This is
the same as requiring that *C* must raise the probability of
*E* in a *weighted average* of background contexts, where
each background context is weighted by the product of \(\PP(B)\) and
the absolute value of

 

 \[\PP(E \mid C \amp B) - \PP(E \mid {\nsim}C \amp B).\]


Dupre's account surely comes closer to capturing our ordinary
use of causal language. Indeed, the inequality in
 Dupre
 is what one looks for in randomized trials. If one randomly
determines which members of a population receive a treatment
(*C*) and which do not \(({\nsim}C)\), then the distribution of
background conditions *B* ought to be the same in both groups,
and ought to reflect the frequency of these conditions in the
population. Thus we would expect the frequency of *E* to be
higher in the treatment group just in case inequality Dupre
holds. 



On the other hand, Eells's population-relative formulation allows
us to make more precise causal claims: in the population as a whole,
smoking is a mixed cause of lung cancer; in the sub-population of
individuals who lack the protective gene, smoking is a positive cause
of lung cancer; in the sub-population consisting of individuals who
possess the gene, smoking is a negative cause of lung cancer.



In any event, this debate does not really seem to be about the
metaphysics of causation. As we saw in the previous section, causal
relevance is really the basic metaphysical concept. The dispute
between Dupre and Eells is really a debate about how best to use the
word 'cause' to pick out a particular species of causal
relevance. Dupre's proposed usage will count as (positive)
causes many things that will be mixed causes in Eells's proposed
usage. But there does not seem to be any underlying disagreement about
which factors are causally relevant. (For defense of a similar
position, see Twardy and Korb 2004.)


### 2.7 Conclusion and Further Reading



The program described in this section did much to illuminate the
relationship between causation and probability. In particular, it
helped us to better understand the way in which causal structure can
give rise to probabilistic relations of screening off. However,
despite the mathematical framework of the program, and points of
contact with statistics and experimental methodology, this program did
not give rise to any new computational tools, or suggest any new
methods for detecting causal relationships. For this reason, the
program has largely been supplanted by the causal modeling tools
described in the next section.



The main works surveyed in this section are Reichenbach 1956, Suppes
1970, Cartwright 1979, Skyrms 1980, and Eells 1991. Williamson 2009
and Hitchcock 2016 are two further surveys that cover a number of the
topics discussed in this section. The entries for
 Hans Reichenbach
 and
 Reichenbach's Common Cause Principle
 include discussions of Reichenbach's program and the status of
his Common Cause Principle. Salmon (1984) contains an extensive
discussion of conjunctive forks. The entry for
 Simpson's paradox
 contains further discussion of some of the issues raised in
 Section 2.4.


## 3. Causal Modeling



The discussion of the previous section conveys some of the complexity
of the problem of inferring causal relationships from probabilistic
correlations. Fairly recently, a number of techniques have been
developed for representing systems of causal relationships, and for
inferring causal relationships from probabilities. The name
'causal modeling' is often used to describe the new
interdisciplinary field devoted to the study of methods of causal
inference. This field includes contributions from statistics,
artificial intelligence, philosophy, econometrics, epidemiology, and
other disciplines. Within this field, the research programs that have
attracted the greatest philosophical interest are those of the
computer scientist Judea Pearl and his collaborators, and of the
philosophers Peter Spirtes, Clark Glymour, and Richard Scheines (SGS)
and their collaborators. The most significant works of these authors
are Pearl (2009) (first published in 2000), and Spirtes et al. (2000)
(first published in 1993).


### 3.1 Graphical Causal Models



Every causal model involves a set of variables \(\bV\). The variables
in \(\bV\) may include, for example, the education-level, income, and
occupation of an individual. A variable could be binary, its values
representing the occurrence or non-occurrence of some event, or the
instantiation or non-instantiation of some property. But as the
example of income suggests, a variable could have multiple values or
even be continuous.



A probabilistic causal model also includes a probability measure P. P
is defined over propositions of the form \(X = x\), where *X* is
a variable in \(\bV\) and *x* is a value in the range of
*X*. P is also defined over conjunctions, disjunctions, and
negations of such propositions. It follows that conditional
probabilities over such propositions will be well-defined whenever the
event conditioned on has positive probability. P is usually understood
to represent some kind of objective probability.



Causal relationships among the variables in \(\bV\) are represented by
*graphs*. We will consider two types of graphs. The first is
the *directed acyclic graph* (DAG). A *directed graph*
\(\bG\) on variable set \(\bV\) is a set of ordered pairs of variables
in \(\bV\). We represent this visually by drawing an arrow from
*X* to *Y* just in case \(\langle X, Y\rangle\) is in
\(\bG\). Figure 3 shows a directed graph on variable set \(\bV = \{S,
T, W, X, Y, Z\}\).



![a diagram with 6 letters. S has an arrow pointing to T which in turn has arrows pointing to X and Y. Y in turn has an arrow pointing to Z which has an arrow pointing to W which has an arrow pointing back to Y.](figure3.svg)

Figure 3




A *path* in a directed graph is a non-repeating sequence of
arrows that have endpoints in common. For example, there is a path
from *X* to *Z*, which we can write as \(X \leftarrow T
\rightarrow Y \rightarrow Z\). A *directed path* is a path in
which all the arrows align by meeting tip-to-tail; for example, there
is a directed path \(S \rightarrow T \rightarrow Y \rightarrow Z\). A
directed graph is *acyclic,* and hence a DAG, if there is no
directed path from a variable to itself. The graph in
 Figure 3
 is a DAG.



The relationships in the graph are often described using the language
of genealogy. The variable *X* is a *parent* of *Y*
just in case there is an arrow directed from *X* to *Y*.
\(\PA(Y)\) will denote the set of all parents of *Y*. In
 Figure 3,
 \(\PA(Y) = \{T, W\}\). *X* is an *ancestor* of *Y*
(and *Y* is a *descendant* of *X*) just in case there
is a directed path from *X* to *Y*. However, it will be
convenient to deviate slightly from the genealogical analogy and
define 'descendant' so that every variable is also
a descendant of itself. \(\DE(X)\) denotes the set of all descendants of
*X*. In
 Figure 3
 \(\DE(T) = \{T, X, Y, Z\}\).



An arrow from *Y* to *Z* in a DAG represents that *Y*
is a *direct cause* of *Z*. Roughly, this means that the
value of *Y* makes some causal difference for the value of
*Z*, and that *Y* influences *Z* through some process
that is not mediated by any other variable in \(\bV\). Directness is
relative to a variable set. We will call the system of direct causal
relations represented in a DAG such as
 Figure 3
 the *causal structure* on the variable set \(\bV\).



A second type of graph that we will consider is an *acyclic
directed mixed graph* (ADMG). An
ADMG, will contain double-headed arrows, as well as single-headed
arrows. A double-headed arrow represents a *latent* common
cause. A latent common cause of variables *X* and *Y* is a
common cause that is not included in the variable set \(\bV\). For
example, suppose that *X* and *Y* share a common cause
*L* (Figure 4(a)). An ADMG on the variable set \(\bV = \{X, Y\}\)
will look like Figure 4(b).





|  |  |  |
| --- | --- | --- |
| a diagram of three letters. L has arrows pointint to X and Y |  | a diagram of two letters. X and Y have a double-headed arrow connecting them. |
| (a) |  | (b) |



Figure 4




We only need to represent missing common causes in this way when they
are *closest* common causes. That is, a graph on \(\bV\) should
contain a double-headed arrow between *X* and *Y* when there
is a variable *L* that is omitted from \(\bV\), such that if
*L* were added to \(\bV\) it would be a *direct cause* of
*X* and *Y*. Double-headed arrows do not give rise to
"genealogical" relationships: in
 Figure 4(b),
 *X* is not a parent, ancestor, or descendant of *Y*.



In an ADMG, we expand the definition of a *path* to include
double-headed arrows. Thus, \(X \leftrightarrow Y\) is a path in the
ADMG shown in
 Figure 4(b).
 *Directed path* retains the same meaning, and a directed path
cannot contain double-headed arrows. An ADMG cannot include a directed
path from a variable to itself.



We will adopt the convention that both DAGs and ADMGs represent the
presence *and absence* of both direct causal relationships and
latent common causes. For example the DAG in
 Figure 3
 represents that *T* is a direct cause of *Y*, that *T*
is not a direct cause of *Z*, and that there are no latent common
causes of any variables.


### 3.2 Identification Problems



We will be interested in a variety of problems that have a general
structure. There will be a *query* concerning some causal
feature of the system being investigated. A query may concern:


* The qualitative causal structure on a set of variables. For
instance, we may ask which DAG, or which ADMG, correctly describes the
causal structure among the variables in \(\bV = \{S, T, W, X, Y,
Z\}\). Or we may be interested in some specific feature of the causal
structure, such as whether *X* is a direct cause of *Y*.
Problems with this type of query are sometimes called *causal
discovery* problems.
* The effects of *interventions*. If we were to intervene in
the causal structure and set the value of *X* to *x*, what
is the probability that *Y* would take the value *y*? For
instance, if we were to intervene by giving drugs to a patient, what
is the probability that he would recover?



A given problem will also have a set of inputs. These fall into a
variety of categories:


* Metaphysical and methodological assumptions: These are general
assumptions about the relationship between causal structure and
probability. These assumptions function like a probabilistic theory of
causation. They tell us, in particular, how causal relationships
constrain probabilistic relationships. The main assumptions that we
will consider are the *Markov Condition* (MC), the
*Minimality Condition*, and the *Faithfulness
Condition.*
* Background assumptions about the causal structure: We may assume
that a particular DAG or ADMG describes the causal structure on some
set of variables, or assume that the correct graph does or does not
include an arrow from *X* to *Y*. In particular, we may have
time order information that constrains the possible causal structures.
For instance, if we know that *X* concerns a time earlier than
*Y*, then we can rule out causal structures in which *Y*
causally influences *X*.
* Observations: Information about the probability distribution P
over the variables in \(\bV\).



In realistic scientific cases, we never directly observe the true
probability distribution P over a set of variables. Rather, we observe
finite data that approximate the true probability when sample sizes
are large enough and observation protocols are well-designed. Since
our primary concern is with the philosophical issue of how
probabilities determine or constrain causal structure, we will not
address these important practical concerns. An answer to a query that
can be determined from the true probabilities is said to be
*identifiable*. For instance, if we can determine the correct
DAG on a variable set \(\bV\) from the probability distribution on
\(\bV\), the DAG is identifiable.


### 3.3 The Markov Condition



The most important principle connecting the causal structure on
\(\bV\), as represented in a graph \(\bG\), and the probability
distribution P on \(\bV\) is the *Markov Condition* (MC). Let
us first consider the case where \(\bG\) is a DAG. Then P satisfies
the *Markov Condition* (MC) relative to \(\bG\) if and only it
satisfies these three conditions:




|  |  |
| --- | --- |
| (MCScreening\_off) | 
For every variable *X* in \(\bV\), and every set of variables
\(\bY \subseteq \bV \setminus \DE(X)\),
\(\PP(X \mid \PA(X) \amp \bY) = \PP(X \mid \PA(X))\).
 |
| (MCFactorization) | 
Let \(\bV = \{X\_1, X\_2 , \ldots ,X\_n\}\). Then
\(\PP(X\_1, X\_2 , \ldots ,X\_n) = \prod\_i \PP(X\_i \mid \PA(X\_i))\).
 |
| (MC*d*-separation) | 
Let \(X, Y \in \bV, \bZ \subseteq \bV \setminus \{X, Y\}\). Then
\(\PP(X, Y \mid \bZ) = \PP(X \mid \bZ) \times \PP(Y \mid
\bZ)\)
if \(\bZ\) *d*-separates *X* and *Y* in \(\bG\)
(explained below).
 |



These three conditions are equivalent when \(\bG\) is a DAG.



Let us take some time to explain each of these formulations.



MCScreening\_off says that the parents of variable *X*
screen *X* off from all other variables, except for the
descendants of *X*. Given the values of the variables that are
parents of *X*, the values of the variables in \(\bY\) (which
includes no descendants of \(X)\), make no further difference to the
probability that *X* will take on any given value.



MCFactorization tells us that once we know the conditional
probability distribution of each variable given its parents, \(\PP(X\_i
\mid \PA(X\_i))\), we can compute the complete joint distribution
over all of the variables. This captures Reichenbach's idea that
probability relations between variables that are not related as cause
and effect are nonetheless derived from probability relations between
causes and effects.



MC*d*-separation uses the graphical notion of
*d*-separation, introduced by Pearl (1988). Let \(X, Y \in \bV,
\bZ \subseteq \bV \setminus \{X, Y\}\). As noted above, a path from
*X* to *Y* is a sequence of variables \(\langle X = X\_1 ,
\ldots ,X\_k = Y\rangle\) such that for each \(X\_i\), \(X\_{i+1}\),
there is either an arrow from \(X\_i\) to \(X\_{i+1}\) or an arrow from
\(X\_{i+1}\) to \(X\_i\) in \(\bG\). A variable \(X\_i , 1 \lt i \lt k\)
is a *collider* on the path just in case there is an arrow from
\(X\_{i-1}\) to \(X\_i\) and from \(X\_{i+1}\) to \(X\_i\). That is,
\(X\_i\) is a collider on a path just in case two arrows converge on
\(X\_i\) in the path. \(\bZ\) *d*-separates *X* and *Y*
just in case every path \(\langle X = X\_1 , \ldots ,X\_k = Y\rangle\)
from *X* to *Y* contains at least one variable \(X\_i\) such
that either: (i) \(X\_i\) is a collider, and no descendant of \(X\_i\)
(including \(X\_i\) itself) is in \(\bZ\); or (ii) \(X\_i\) is not a
collider, and \(X\_i\) is in \(\bZ\). MC*d*-separation
states that *d*-separation is sufficient for conditional
independence.



Note that MC provides sufficient conditions for variables to be
probabilistically independent, conditional on others, but no necessary
condition. The Markov Condition entails many of the same screening off
relations as Reichenbach's Common Cause Principle, discussed in
 Section 2.3
 above. Here are some examples:



![a diagram of 5 letters. T has an arrow to W which has arrows to X and Z. X in turn has an arrow to Y.](figure5.svg)

Figure 5




In Figure 5, MC implies that *X* screens *Y* off from all of
the other variables, and that *W* screens *Z* off from all
of the other variables. This is most easily seen from MCScreening
off. *W* also screens *T* off from all of the other
variables, which is most easily seen from
MC*d*-separation. MC does not imply that *T*
screens *Y* off from *Z* (or indeed anything from anything).
While *Y* and *Z* do have a common cause that screens them
off (*W*), not all common causes screen them off (*T* does
not have to), and not everything that screens them off is a common
cause (*X* screens them off but is not a common cause).



![a diagram of 3 letters. X and Y both have arrows pointing to Z.](figure6.svg)

Figure 6




In Figure 6, MC entails that *X* and *Y* will be
unconditionally independent, but not that they will be independent
conditional on *Z*. This is most easily seen from
MC*d*-separation.



MC is not expected to hold for arbitrary sets of variables \(\bV\),
even when the graph \(\bG\) accurately represents the causal relations
among those variables. For example, MC will typically fail in the
following kinds of case:


1. In an EPR (Einstein-Podolsky-Rosen) set-up, we have two particles
prepared in the singlet state. If *X* represents a spin
measurement on one particle, *Y* a spin measurement (in the same
direction) on the other, then *X* and *Y* are perfectly
anti-correlated. (One particle will be spin-up just in case the other
is spin-down.) The measurements can be conducted sufficiently far away
from each other that it is impossible for one outcome to causally
influence the other. However, it can be shown that there is no (local)
common cause *Z* that screens off the two measurement outcomes.
2. The variables in \(\bV\) are not appropriately distinct. For
example, suppose that *X*, *Y*, and *Z* are variables
that are probabilistically independent and causally unrelated. Now
define \(U = X + Y\) and \(W = Y + Z\), and let \(\bV = \{U, W\}\).
Then *U* and *W* will be probabilistically dependent, even
though there is no causal relation between them.
3. MC may fail if the variables are too coarsely grained. Suppose
*X*, *Y*, and *Z* are quantitative variables, *Z*
is a common cause of *X* and *Y*, and neither *X* nor
*Y* causes the other. Suppose we replace *Z* with a coarser
variable, \(Z'\) indicating only whether *Z* is high or low. Then
we would not expect \(Z'\) to screen *X* off from *Y*. The
value of *X* may well contain information about the value of
*Z* beyond what is given by \(Z'\), and this may affect the
probability of *Y*.



If there are latent common causes, we expect MCScreening
off and MCFactorization to fail if we apply them in a
naive way. For example, suppose that the true causal structure on
\(\bV = \{X, Y, Z\}\) is shown by the ADMG in Figure 7.



![a diagram of three letters. X has an arrow to its right pointing to Y which in turn has an arrow to its right to Z; a double headed curved arrow connects X and Z. ](figure7.svg)

Figure 7




*Y* is the only parent of *Z* shown in the graph, and if we
try to apply MCScreening\_off, it tells us that *Y*
should screen *X* off from *Z*. However, we would expect
*X* and *Z* to be correlated, even when we condition on
*Y*, due to the latent common cause. The problem is that the
graph is missing a relevant parent of *Z*, namely the omitted
common cause. However, suppose that the probability distribution is
such that *if* the latent cause *L* were added, the
probability distribution over the expanded set of variables would
satisfy MC with respect to the resulting DAG. Then it turns out that
the probability distribution will *still* satisfy
MC*d*-separation with respect to the ADMG of
 Figure 8.
 This requires us to expand the definition of *d*-separation to
include paths with double-headed arrows. For instance, *Z* is a
collider on the path \(Y \rightarrow Z \leftrightarrow X\) (since
*Z* has two arrows pointing into it), but *X* is not a
collider on the path \(Y \leftarrow X \leftrightarrow Z\). Thus we
will say that a probability distribution P satisfies the Markov
Condition relative to an ADMG just in case it satisfies
MC*d*-separation.



Both SGS 2000 and Pearl 2009 contain statements of a principle called
the *Causal Markov Condition* (CMC), but they mean different
things. In Pearl's formulation, CMC is just a statement of a
mathematical theorem: Pearl and Verma (1991) prove if each variable in
\(\bV\) is a deterministic product of its parents in \(\bV\), together
with an error term; and the errors are probabilistically independent
of each other; then the probability distribution on \(\bV\) will
satisfy MC with respect to the DAG \(\bG\). Pearl interprets this
result in the following way: Macroscopic systems, he believes, are
deterministic. In practice, however, we never have access to all of
the causally relevant variables affecting a macroscopic system. But if
we include enough variables in our model so that the excluded
variables are probabilistically independent of one another, then our
model will satisfy the MC, and we will have a powerful set of analytic
tools for studying the system. Thus MC characterizes a point at which
we have constructed a useful approximation of the complete system.



In SGS 2000, the CMC has more the status of an empirical posit. If
\(\bV\) is set of macroscopic variables that are well-chosen, meaning
that they are free from the sorts of defects described in points (ii)
and (iii) above; \(\bG\) is a graph representing the causal structure
on \(\bV\); and P is the objective probability distribution resulting
from this causal structure; then P can be expected to satisfy MC
relative to \(\bG\). More precisely, P will satisfy all three versions
of MC if \(\bG\) is a directed acyclic graph, and P will satisfy
MC*d*-separation if \(\bG\) is an ADMG with
double-headed arrows. SGS defend this empirical posit in two different
ways:


1. Empirically, it seems that a great many systems do in fact satisfy
MC.
2. Many of the methods that are in fact used to detect causal
relationships tacitly presuppose the MC. In particular, the use of
randomized trials presupposes a special case of the MC. Suppose that
an experimenter determines randomly which subjects will receive
treatment with a drug \((D = 1)\) and which will receive a placebo
\((D = 0)\), and that under this regimen, treatment is
probabilistically correlated with recovery (*R*). The effect of
randomization is to eliminate all of the parents of *D*, so MC
tells us that if *R* is not a descendant of *D*, then
*R* and *D* should be probabilistically independent. If we
do not make this assumption, how can we infer from the experiment that
*D* is a cause of *R*?



Cartwright (1993, 2007: chapter 8) has argued that MC need not hold
for genuinely indeterministic systems. Hausman and Woodward (1999,
2004) attempt to defend MC for indeterministic systems.



A causal model that comprises a DAG and a probability distribution
that satisfies MC is called a *causal Bayes net* (CBN). A
causal model incorporating an ADMG and probability distribution
satisfying MC*d*-separation is called a
*semi-Markov causal model* (SMCM).


### 3.4 The Minimality and Faithfulness Conditions



The MC states a sufficient condition but not a necessary condition for
conditional probabilistic independence. As such, the MC by itself can
never entail that two variables are conditionally or unconditionally
dependent. The Minimality and Faithfulness Conditions are two
principles that posit necessary conditions for probabilistic
independence. The terminology comes from Spirtes et al. (2000). Pearl
provides analogous conditions with different terminology.



(i) *The Minimality Condition*. Suppose that the acyclic
directed graph \(\bG\) on variable set \(\bV\) satisfies MC with
respect to the probability distribution P. The Minimality Condition
asserts that no sub-graph of \(\bG\) over \(\bV\) also satisfies the
Markov Condition with respect to P. (A subgraph of \(\bG\) is a graph
over \(\bV\) that results from removing arrows from \(\bG)\). As an
illustration, consider the variable set \(\{X, Y\}\), let there be an
arrow from *X* to *Y*, and suppose that *X* and
*Y* are probabilistically independent of each other according to
probability function P. This graph would satisfy the MC with respect
to P: none of the independence relations mandated by the MC are absent
(in fact, the MC mandates no independence relations). But this graph
would violate the Minimality Condition with respect to P, since the
subgraph that omits the arrow from *X* to *Y* would also
satisfy the MC. The Minimality Condition implies that if there is an
arrow from *X* to *Y*, then *X* makes a probabilistic
difference for *Y*, conditional on the other parents of *Y*.
In other words, if \(\bZ = \PA(Y) \setminus \{X\}\), there exist
\(\bz\), *y*, *x*, \(x'\) such that

 

 \[\PP(Y = y \mid X = x \amp \bZ = \bz) \ne \PP(Y = y \mid X = x' \amp \bZ = \bz).\]


(ii) *The Faithfulness Condition*. The Faithfulness Condition
says that all of the (conditional and unconditional) probabilistic
independencies that exist among the variables in \(\bV\) are
*required* by the MC. For example, suppose that \(\bV = \{X, Y,
Z\}\). Suppose also that *X* and *Y* are unconditionally
independent of one another, but dependent, conditional upon *Z*.
(The other two variable pairs are dependent, both conditionally and
unconditionally.) The graph shown in
 Figure 8
 does not satisfy the faithfulness condition with respect to this
distribution (colloquially, the graph is not faithful to the
distribution). MC, when applied to the graph of
 Figure 8,
 does not imply the independence of *X* and *Y*. By
contrast, the graph shown in
 Figure 6
 above is faithful to the described distribution. Note that
 Figure 8
 does satisfy the Minimality Condition with respect to the
distribution; no subgraph satisfies MC with respect to the described
distribution. In fact, the Faithfulness Condition is strictly stronger
than the Minimality Condition.



![a diagram of three letters with X having arrows pointing to Z and Y and also an arrow from Z to Y.](figure8.svg)

Figure 8




The Faithfulness Condition implies that the causal influences of one
variable on another along multiple causal routes do not
'cancel'. In Figure 8, *X* influences *Y* along
two different directed paths. If the effect of one path is to exactly
undo the influence along the other path, then *X* and *Y*
will be probabilistically independent. The Faithfulness Condition
forbids such exact cancellation. This 'no canceling'
condition seems implausible as a metaphysical or conceptual constraint
upon the connection between causation and probabilities. For example,
if one gene codes for the production of a particular protein, and
suppresses another gene that codes for the same protein, the operation
of the first gene will be independent of the presence of the protein.
Cartwright (2007: chapter 6) and Andersen (2103) argue that violations
of faithfulness are widespread.



The Faithfulness Condition is a *methodological* principle
rather than a metaphysical principle. Given a distribution on \(\{X,
Y, Z\}\) in which *X* and *Y* are independent, we should
infer that the causal structure is that depicted in
 Figure 6,
 rather than
 Figure 8.
 This is not because
 Figure 8
 is conclusively ruled out by the distribution, but rather because it
is preferable to postulate a causal structure that *implies*
the independence of *X* and *Y* rather than one that is
merely *consistent* with independence.


### 3.5 Identifiability of Causal Structure



The original hope of Reichenbach and Suppes was to provide a reduction
of causation to probabilities. To what extent has this hope been
realized within the causal modeling framework? Causal modeling does
not offer a reduction in the traditional philosophical sense; that is,
it does not offer an analysis of the form '*X* causes
*Y* if and only if...' where the right hand side of
the bi-conditional makes no reference to causation. Instead, it offers
a series of postulates about how causal structure constrains the
values of probabilities. Still, if we have a set of variables \(\bV\)
and a probability distribution P on \(\bV\), we may ask if P suffices
to pick out a unique causal graph \(\bG\) on \(\bV\).



Pearl (1988: Chapter 3) proves the following theorem:



(Identifiability with time-order)


If


1. the variables in \(\bV\) are time-indexed, such that only earlier
variables can cause later ones;
2. the probability P assigns positive probability to every possible
assignment of values of the variables in \(\bV\);
3. there are no latent common causes, so that the correct causal
graph \(\bG\) is a DAG;
4. and the probability measure P satisfies the Markov and Minimality
Conditions with respect to \(\bG\);



then it will be possible to uniquely identify \(\bG\) on the basis of
P.





In many ways, this result successfully executes the sort of project
described in Section 2
above. That is, making the same sorts of assumptions about
time-indexing, and substantive assumptions about the connection
between probability and causation, it establishes that it is possible
to identify causal structure using probabilities.



If we don't have information about time ordering, or other
substantive assumptions restricting the possible causal structures
among the variables in \(\bV\), then it will not always be possible to
identify the causal structure from probability alone. In general,
given a probability distribution P on \(\bV\), it is only possible to
identify a *Markov equivalence class* of causal structures.
This will be the set of all DAGs on \(\bV\) that (together with MC)
imply all and only the conditional independence relations contained in
P. The *PC algorithm* (SGS 2000: 84-85), named for its
two creators (*P*eter Spirtes and *C*lark Glymour), is one
algorithm that generates the Markov equivalence class for any given
probability distribution.




Consider two simple examples involving three variables \(\{X, Y,
Z\}\). Suppose our probability distribution has the following
properties:


* *X* and *Y* are dependent unconditionally, and
conditional on *Z*
* *Y* and *Z* are dependent unconditionally, and
conditional on *X*
* *X* and *Z* are dependent unconditionally, but
independent conditional on *Y*



Then the Markov equivalence class is:

 

\[
\begin{align}
X \rightarrow Y \rightarrow Z\\
X \leftarrow Y \leftarrow Z\\
X \leftarrow Y \rightarrow Z
\end{align}
\]


We cannot determine from the probability distribution, together with
MC and Faithfulness, which of these structures is correct.



On the other hand, suppose the probability distribution is as
follows:


* *X* and *Y* are dependent unconditionally, and
conditional on *Z*
* *Y* and *Z* are dependent unconditionally, and
conditional on *X*
* *X* and *Z* are independent unconditionally, but
dependent conditional on *Y*



Then the Markov equivalence class is:

 

\[X \rightarrow Y \leftarrow Z\]


Note that the first probability distribution on \(\{X, Y, Z\}\) is
that characterized by Reichenbach's Common Cause Principle. The
second distribution reverses the relations between *X* and
*Z*: they are unconditionally *independent* and
conditionally *dependent*. Contrary to Reichenbach, it is
actually the latter pattern of dependence relations that is most
useful for orienting the causal arrows in the graph. In the last
causal structure shown, *Y* is a collider on the path from
*X* to *Z*. MC*d*-separation implies that
colliders give rise to distinctive conditional independence relations,
while all three types of non-collider give rise to the same
conditional independence relations. Many of the algorithms that have
been developed for inferring causal structure from probabilities work
by searching for colliders (see, e.g., SGS 2000: Chapter 5).



The identifiability results discussed so far all assume that the
correct causal graph is a DAG. However, it is common that latent
variables will be present, and even more common that we might wish to
allow for the possibility of latent variables (whether they are
actually there or not). If we allow that the correct causal graph may
contain double-headed arrows, we can still apply
MC*d*-separation, and ask which graphs imply the same
sets of conditional independence relations. The
Markov equivalence class will be larger than it was when we did not
allow for latent variables. For instance, given the last set of
probability relations described above, the graph

 

\[X \rightarrow Y \leftarrow Z\]


is no longer the only one compatible with this distribution. The
structure

 

\[X \leftrightarrow Y \leftrightarrow Z\]


is also possible, as are several others.


### 3.6 Interventions



A conditional probability such as \(\PP(Y = y \mid X = x)\) gives
us the probability that *Y* will take the value *y*, given
that *X* has been *observed* to take the value *x*.
Often, however, we are interested in predicting the value of *Y*
that will result if we *intervene* to set the value of *X*
equal to some particular value *x*. Pearl writes \(\PP(Y = y
\mid \do(X = x))\) to characterize this probability. What is the
difference between observation and intervention? When we merely
observe the value that a variable takes, we are learning about the
value of the variable when it is caused in the normal way, as
represented in our causal model. Information about the value of the
variable will also provide us with information about its causes, and
about other effects of those causes. However, when we intervene, we
override the normal causal structure, forcing a variable to take a
value it might not have taken if the system were left alone. The value
of the variable is determined completely by our intervention, the
causal influence of the other variables being completely overridden.
Graphically, we can represent the effect of this intervention by
eliminating the arrows directed into the variables intervened upon.
Such an intervention is sometimes described as 'breaking'
those arrows. 



A causal model can be used to predict the effects of such an
intervention. Suppose we have a causal model in which the probability
distribution P satisfies MC on the causal DAG \(\bG\) over the
variable set \(\bV = \{X\_1, X\_2 ,\ldots ,X\_n\}\). The most useful
version of MC for thinking about interventions is
MCFactorization (see
 Section 3.3),
 which tells us:

 

\[\PP(X\_1, X\_2 , \ldots ,X\_n) = \prod\_i \PP(X\_i \mid \PA(X\_i))\]


Now suppose that we intervene by setting the value of \(X\_k\) to
\(x\_k\). The post-intervention probability \(\PP'\) is the result of
altering the factorization as follows:

 

\[\PP'(X\_1, X\_2 , \ldots ,X\_n) = \PP'(X\_k) \times \prod\_{i\ne k} \PP(X\_i \mid \PA(X\_i)),\]


where \(\PP'(X\_k = x\_k) = 1\). The conditional probabilities of the
form \(\PP(X\_i \mid \PA(X\_i))\) for \(i \ne k\) remain unchanged
by the intervention.



This treatment of interventions has been expanded in a number of
directions. The 'manipulation theorem' (theorem 3.6 of SGS
2000) generalizes the formula to cover a much broader class of
interventions, including ones that don't break all the arrows
into the variables that are intervened on. Pearl (2009: Chapter 3)
develops an axiomatic system he calls the 'do-calculus'
for computing post-intervention probabilities that can be applied to
systems with latent variables. 


### 3.7 Conclusion and Further Reading



Causal modeling is a burgeoning area of research. This entry has largely
ignored work on computational methods, as well as applications of the
tools discussed here. Rather, the focus has been on the conceptual
underpinnings of recent programs in causal modeling, with special
attention to the connection between causation and probability. It has
also focused on what it possible to learn about causation "in
principle" on the basis of probabilities, while ignoring the
practical problems of making causal inferences on the basis of finite
data samples (which inevitably deviate from the true
probabilities).



The entry on Causal Models covers all of the material in this section
in greater detail. The most important works surveyed in this section
are Pearl 2009 and Spirtes, Glymour, & Scheines 2000. Pearl 2010
is a short overview of Pearl's program, and Pearl et al. 2016 is
a longer overview. The latter, in particular, assumes relatively
little technical background. Scheines 1997 and the Introduction of
Glymour & Cooper 1999 are accessible introductions to the SGS
program. Neapolitan 2004 is a text book that treats Bayes nets in
causal and noncausal contexts. Neapolitan & Jiang 2016 is a short
overview of this topic. Hausman 1999, Glymour 2009, Hitchcock 2009,
and Eberhardt 2017 are short overviews that cover some of the topics
raised in this section. The entry on
 causation and manipulability
 contains extensive discussion of interventions, and some discussion
of causal models. 


## 4. Actual Causation



Many philosophers and legal theorists have been interested in the
relation of *actual causation*. This concerns the assignment of
causal responsibility for an event, based on how events actually play
out. For example, suppose that Billy and Suzy each throw a rock at a
bottle, and that each has a certain probability of hitting and
breaking it. As it happens, Suzy's rock hits the bottle, and
Billy's doesn't. As things actually happened, we would say
that Suzy's throw caused the bottle to shatter, while
Billy's didn't. Nonetheless, Billy's throw increased
the probability that the bottle would shatter, and it would be
identified as a cause by the theories described in sections
 2
 and
 3.
 Billy's throw had a tendency to shatter the bottle; it was a
potential cause of the bottle shattering; it was the sort of thing
that generally causes shattering; but it did not actually cause the
bottle to shatter. 



A number of authors have attempted to provide probabilistic analyses
of actual causation. Some, such as Eells (1991: chapter 6), Kvart
(1997, 2004), and Glynn
(2011), pay careful attention to the way in which probabilities change
over time. Some, such as Dowe (2004) and Schaffer (2001), combine
probabilities with the resources of a process theory of causation.
Some, such as Lewis (1986b), Menzies (1989), and Noordhof (1999),
employ probabilities together with counterfactuals to analyze actual
causation. And others such as Beckers & Vennekens (2016),
Fenton-Glynn (2017), Halpern (2016: Section 2.5), Hitchcock (2004a),
and Twardy & Korb (2011) employ causal modeling tools similar to
those described in
 Section 3.
 We will describe two of those theories--Lewis (1986b) and
Fenton-Glynn (2017)--in more detail in sections
 4.3
 and
 4.4
 below. 


### 4.1 A First Attempt



In
 Section 2.5
 above, we saw that Eells (1991) defines a variety of different ways
in which *C* can be causally relevant for *E*. *C* can
be a positive, negative, or mixed cause of *E* depending upon
whether *C* raises, lowers, or leaves unchanged the probability
of *E* in various background conditions \(B\_i\). A natural
suggestion is that (i) an actual cause of *E* is a type of
positive cause of *E*; but (ii) for assessing actual causation,
only the background condition that *actually* obtains is
relevant. Putting these ideas together, we get:



(AC1)\(C\_t\) is an
actual cause of \(E\_{t'}\) just in case:

(i)\(t \lt t'\)
(ii)\(\PP(E\_{t'} \mid
 C\_t \amp B\_a) \gt \PP(E\_{t'} \mid {\nsim}C\_t \amp B\_a)\) , where
 \(B\_a\) is the background condition that actually obtains


As we shall see in the next section, this type of analysis is
vulnerable to two types of counterexamples: cases where causes seem to
lower (or leave unchanged) the probabilities of their effects; and
cases where non-causes seem to raise the probabilities of events that
are not their effects. Most of the theories mentioned in the previous
section can be seen as attempts to improve upon AC1 to deal with these
types of counterexample.


### 4.2 Problem Cases



Actual causes can sometimes lower the probability of their effects in
cases of *preemption*: Suppose that Billy and Suzy are aiming
rocks at a bottle. Billy decides that he will give Suzy the
opportunity to throw first; he will throw his rock just in case Suzy
doesn't throw hers. For mathematical convenience, we will assume
that there is some small probability--0.1 say--that Billy
does not faithfully execute his plan. Billy is a more accurate thrower
than Suzy. If Billy throws his rock, there is a 90% chance that it
will shatter the bottle; if Suzy throws, she has a 50% chance of
success. Suzy throws her rock and Billy doesn't; Suzy's
rock hits the bottle and smashes it. By throwing, Suzy lowered the
probability of shattering from 81% (the probability that Billy would
both throw and hit if Suzy hadn't thrown) to 54.5%
(accommodating the small probability that Billy will throw even if
Suzy throws). Suzy's throw *preempts* Billy's
throw: she prevents Billy from throwing, and substitutes her own, less
reliable throw. Nonetheless, Suzy's throw actually caused the
bottle to shatter.



Changing the example slightly gives us a case of a probability-raising
non-cause. Suppose that Billy and Suzy throw their rocks
simultaneously. As it happens, Suzy's throw hits the bottle and
Billy's misses. Nonetheless, Billy's throw increased the
probability that the bottle would shatter from 50% (the probability
that Suzy would hit) to 95% (the probability that at least one of them
would hit). But Billy's throw did not in fact cause the bottle
to shatter. In the terminology of Schaffer (2001), Billy's throw
is a *fizzler*. It had the potential to shatter the bottle, but
it fizzled out, and something else actually caused the bottle to
break.


### 4.3 Lewis's Counterfactual Theory



David Lewis is the best-known advocate of a counterfactual theory of
causation. In Lewis 1973, he offered a counterfactual theory of
causation under the assumption of determinism. Lewis 1986b presented a
probabilistic extension to this counterfactual theory of
causation.



Lewis defines a relation of *causal dependence* that is
sufficient, but not necessary for causation.



(CD)Event *E* *causally depends upon* event 
 *C* just in case:

(i)*C* and *E* actually occur, at times *t*
 and \(t'\) respectively.
(ii)At time *t*, the probability of *E* was 
 *x*.
(iii)If *C* hadn't occurred, then at time 
 *t* the probability of *E* would have been less than or 
 equal to *y*.
(iv)\(x \gg y\).


The counterfactual in (iii) is to be understood in terms of possible
worlds: it says that in the nearest possible world(s) where *C*
does not occur, the probability of *E* is less than or equal to
*y*. (There needn't be a single value that the probability
would have been. It can take on different values in the closest
possible worlds, as long as all of those values are less than or equal
to *y*.) On this account, the relevant notion of
'probability-raising' is not understood in terms of
conditional probabilities, but in terms of unconditional probabilities
in different possible worlds.



Lewis defines causation (what we are calling "actual
causation") to be the *ancestral* of causal dependence;
that is:



(Lewis)*C* causes
*E* just in case there is a sequence of events \(D\_1\), \(D\_2\),
..., \(D\_n\), such that \(D\_1\) causally depends upon \(C\),
\(D\_2\) causally depends upon \(D\_1\), ..., *E* causally
depends upon \(D\_n\).

This definition guarantees that causation will be transitive: if
*C* causes *D*, and *D* causes *E*, then *C*
causes *E*. This modification is useful for addressing certain
types of preemption. Consider the example from the previous section,
where Suzy throws her rock, preempting Billy. We can interpolate an
event *D* between Suzy's throw, *C*, and the
bottle's shattering *E*. Let *D* be the presence of
Suzy's rock on its actual trajectory, at some time after Billy
has already failed to throw. If Suzy hadn't thrown, *D*
would have been much less likely. And if *D* hadn't
occurred, *E* would have been much less probable. Since *D*
occurs after Billy has already declined to throw, if *D*
hadn't occurred, there would not have been *any* rock on
a trajectory toward the bottle. Thus there is a chain of causal
dependence from *C* to *D* to *E*. 



Despite this success, it has been widely acknowledged (even by Lewis
himself) that Lewis's probabilistic theory has problems with
other types of preemption, and with probability-raising
non-causes.


### 4.4. Fenton-Glynn's Causal Modeling Account



Fenton-Glynn (2017) offers an analysis of actual causation that is
based on the definition of Halpern and Pearl (2005), who consider only
the deterministic case. What follows here is a simplified version of
Fenton-Glynn's proposal, as one example of an analysis employing
causal models.



Let \(\bV\) be a set of time-indexed, binary variables, which we
assume to include any common causes of variables in \(\bV\) (so that
the correct causal graph on \(\bV\) is a DAG). Let \(\*\) be an assignment
function that assigns to each variable \(X\) in \(\bV\) one of its
possible values. Intuitively, \(\*\) identifies the *actual* value
of each variable. We will denote \(\*(X)\) by \(x^\*\), and \(x'\)
will denote the non-actual value of \(X\). If \(\bX\) is a set of
variables in \(\bV\), \(\bX\) = \(\bx^\*\) will be a proposition stating that
each variable in \(\bX\) takes the actual value assigned by \(\*\). Let P
be a probability function on \(\bV\) representing objective
probability, which we assume to satisfy the Markov and Minimality
Conditions (Sections
 3.3
 and
 3.4
 above). We also assume that P assigns positive probability to every
possible assignment of values to variables in \(\bV\).



Given the identifiability result described in
 Section 3.5
 above, we can recover the correct causal graph \(\bG\) from the probability
function P together with the time-indices of the variables. We can now
use P and \(\bG\) to compute the effects of interventions, as described in
 section 3.6
 above. We now define actual causation as follows:



(F-G)Let \(X, Y \in \bV\). \(X = x^\*\) is an actual cause 
 of \(Y = y^\*\) just in case there is a partition \(\{\bZ, \bW\}\) 
 of the variables in \(\bV \setminus \{X, Y\}\) such that for every 
 \(\bU \subseteq \bZ\): 

\[\begin{align}
\PP(Y = y^\* &\mid do(X = x^\*, \bW = \bw^\*, \bU = \bu^\*))\\
 \mathbin{\gt} &\PP(Y = y^\* \mid do(X = x',\bW = \bw^\*))\\
\end{align}\]


Intuitively, this is what is going on: If \(X = x^\*\) is an actual
cause of \(Y = y^\*\) then there has to be at least one directed path
from \(X\) to \(Y\). \(\bZ\) will consist of variables that lie
along some (but not necessarily all) of these paths. (If \(X\) is a
direct cause of \(Y\), then \(\bZ\) can be empty.). F-G requires
that \(X = x^\*\) raises the probability of \(Y = y^\*\) in the sense that
interventions that set \(X\) to \(x^\*\) result in higher
probabilities for \(Y = y^\*\) than interventions that set \(X\) to
\(x'\). Specifically, \(X = x^\*\) must raise the probability of \(Y =
y^\*\) when we also intervene to set the variables in \(\bW\) to their
actual values. \(\bW = \bw^\*\) is like a background context of the sort
discussed in
 Section 2.4,
 except that \(\bW\) may include some variables that are descendants
of \(X\). Moreover, \(X = x^\*\) must raise the probability of \(Y =
y^\*\) in conjunction with any combination of variables in \(\bZ\) being
set to their actual values. The idea is that the probabilistic impact
of \(X\) on \(Y\) is being constrained to flow through the
variables in \(\bZ\), and at every stage in the process, the value of
the variables in \(\{X\} \cup \bZ\) must confer a higher probability
on \(Y = y^\*\) than the baseline probability that would have resulted
if \(X\) had been set to \(x'\).



Let's see how this account handles the problem cases from
 section 4.2.
 For the example of preemption, we will use the following
variables:


* \(\ST\_0= 1\) if Suzy throws, 0 if not
* \(\BT\_1= 1\) if Billy throws, 0 if not
* \(\BS\_2 = 1\) if the bottle shatters, 0 if not



The subscripts indicate the relative times of the events, with larger
numbers corresponding to later times. The actual values of the
variables are \(\ST\_0= 1\), \(\BT\_1= 0\), and \(\BS\_2= 1\). The
probabilities are:

 

\[\begin{align}
\PP(\BT\_1= 1 \mid \ST\_0= 1) &{} = .1 \\
\PP(\BT\_1= 1 \mid \ST\_0= 0) &{} = .9 \\[1ex]

\PP(\BS\_2= 1 \mid \ST\_0= 1 \amp \BT\_1= 1) &{} = .95\\
\PP(\BS\_2= 1 \mid \ST\_0= 1 \amp \BT\_1= 0) &{} = .5\\
\PP(\BS\_2= 1 \mid \ST\_0= 0 \amp \BT\_1= 1) &{} = .9\\
\PP(\BS\_2= 1 \mid \ST\_0= 0 \amp \BT\_1= 0) &{} = .01\\
\end{align}\]


(Note that we have added a small probability for the bottle to shatter
due to some other cause, even if neither Suzy nor Billy throw their
rock. This ensures that the probabilities of all assignments of values
to the variables are positive.) The corresponding graph is shown in
Figure 9.



![a three node diagram with node ST_0 having arrows pointing to nodes BS_2 and BT_1. Node BT_1 also has an arrow pointing to node BS_2.](figure9.svg)

Figure 9




Applying F-G, we can take \(\bW = \{\BT\_1\}\), \(\bZ = \varnothing\).
We have:

 

\[\begin{align}
\PP(\BS\_2= 1 \mid \do(\ST\_0= 1) \amp \do(\BT\_1= 0)) &{} = .5\\
\PP(\BS\_2= 1 \mid \do(\ST\_0= 0) \amp \do(\BT\_1= 0)) &{} = .01\\
\end{align}\]


Holding fixed that Billy doesn't throw, Suzy's throw
raises the probability that the bottle will shatter. Thus the
conditions are met for \(\ST = 1\) to be an actual cause of \(\BS =
1\).



To treat the case of fizzling from
 Section 4.2,
 let


* \(\ST\_0= 1\) if Suzy throws, 0 if not
* \(\BT\_0= 1\) if Billy throws, 0 if not
* \(\SH\_1= 1\) if Suzy's rock hits the bottle, 0 if not
* \(\BH\_1= 1\) if Billy's rock hits the bottle, 0 if not
* \(\BS\_2= 1\) if the bottle shatters, 0 if not



The actual values are \(\ST\_0= 1\), \(\BT\_0= 1\), \(\SH\_1= 1\),
\(\BH\_1= 0\), and \(\BS\_2= 1\). The probabilities are:



\[\begin{align}
\PP(\SH\_1= 1 \mid \ST\_0= 1) &{} = .5\\
\PP(\SH\_1= 1 \mid \ST\_0= 0) &{} = .01\\[2ex]

\PP(\BH\_1= 1 \mid \BT\_0= 1) &{} = .9\\
\PP(\BH\_1= 1 \mid \BT\_0= 0) &{} = .01\\[2ex]

\PP(\BS\_2= 1 \mid \SH\_1= 1 \amp \BH\_1= 1) & {} = .998 \\
\PP(\BS\_2= 1 \mid \SH\_1= 1 \amp \BH\_1= 0) & {} = .95\\
\PP(\BS\_2= 1 \mid \SH\_1= 0 \amp \BH\_1= 1) & {} = .95 \\
\PP(\BS\_2= 1 \mid \SH\_1= 0 \amp \BH\_1= 0) & {} = .01\\
\end{align}\]


As before, we have assigned probabilities close to, but not equal to,
zero and one for some of the possibilities. The graph is shown in
Figure 10.



![Five node diagram with node ST_0 pointing to node SH_1 which points to node BS_2. Node BT_0 points to BH_1 which points to the aforementioned node BS_2.](figure10.svg)

Figure 10




We want to show that \(\BT\_0= 1\) is not an actual cause of \(\BS\_2=
1\) according to F-G. We will show this by means of a dilemma: is
\(\BH\_1\in \bW\) or is \(\BH\_1\in \bZ\)?



Suppose first that \(\BH\_1\in \bW\). Then, regardless of whether
\(\ST\_0\) and \(\SH\_1\) are in \(\bW\) or \(\bZ\), we will need to
have 

 

\[\begin{align}
\PP(\BS\_2 = 1 &\mid do(\BT\_0= 1, \BH\_1= 0, \ST\_0= 1, \SH\_1= 1))\\
 \mathbin{\gt} &\PP(\BS\_2 = 1 \mid do(\BT\_0= 0,\BH\_1= 0, \ST\_0= 1, \SH\_1= 1))\\
\end{align}\]


But in fact both of these probabilities are equal to .95. If we
intervene to set \(\BH\_1\) to 0, intervening on \(\BT\_0\) makes no
difference to the probability of \(\BS\_2= 1\).



So let us suppose instead that \(\BH\_1\in \bZ\). Then we will need to
have 

 


\[\begin{align}
\PP(\BS\_2 = 1 &\mid do(\BT\_0= 1, \BH\_1= 0, \ST\_0= 1, \SH\_1= 1))\\
 \mathbin{\gt} &\PP(\BS\_2 = 1 \mid do(\BT\_0= 0, \ST\_0= 1, \SH\_1= 1))\\
\end{align}\]


This inequality is slightly different, since \(\BH\_1= 0\) does
not appear in the second conditional probability. Nonetheless we have


 

\[
\PP(\BS\_2 = 1 \mid do(\BT\_0= 1, \BH\_1= 0, \ST\_0= 1, \SH\_1= 1)) = .95
\]


and

 

\[
\PP(\BS\_2 = 1 \mid do(\BT\_0= 0, \ST\_0= 1, \SH\_1= 1)) = .9505
\]


(The second probability is a tiny bit larger, due to the very small
probability that Billy's rock will hit even if he doesn't
throw it.)



So regardless of whether \(\BH\_1\in \bW\) or is \(\BH\_1\in \bZ\),
condition F-G is not satisfied, and \(\BT\_0= 1\) is not judged to be
an actual cause of \(\BS\_2= 1\). The key idea is that it is not enough
for Billy's throw to raise the probability of the bottle
shattering; Billy's throw together with what happens afterwards
has to raise the probability of shattering. As things actually
happened, Billy's rock missed the bottle. Billy's throw
*together with* his rock missing does not raise the probability
of shattering.



Note that this treatment of fizzling requires that we include
variables for whether the rocks hit the bottle. If we try to model
this case using just three variables, \(\BT\), \(\ST\), and \(\BS\),
we will incorrectly judge that Billy's throw is a cause of the
bottle shattering. This raises the question of what is the
"right" model to use, and whether we can know if we have
included "enough" variables in our model. Fenton-Glynn
(2017) includes some discussion of these tricky issues.


### 4.5 Conclusion and Further Reading



While this section describes some success stories, it is safe to say
that no analysis of actual causation is widely believed to perfectly
capture all of our pre-theoretic intuitions about hypothetical cases.
Indeed, it is not clear that these intuitions form a coherent set, or
that they are perfectly tracking objective features of the world.
Glymour et al. (2010) raise a number of challenges to the general
project of trying to provide an analysis of actual causation.



The anthologies Collins et al. 2004 and Dowe & Noordhof 2004
contain a number of essays on topics related to the discussion of this
section. Hitchcock 2004b has an extended discussion of the problem
posed by fizzlers. Hitchcock 2015 is an overview of Lewis's work
on causation. The entry for
 counterfactual theories of causation
 discusses Lewis's work, and counterfactual theories of
causation more generally.