## 1. Introduction



Causal modeling is an interdisciplinary field that has its origin in
the statistical revolution of the 1920s, especially in the work of the
American biologist and statistician Sewall Wright (1921). Important
contributions have come from computer science, econometrics,
epidemiology, philosophy, statistics, and other disciplines. Given the
importance of causation to many areas of philosophy, there has been
growing philosophical interest in the use of mathematical causal
models. Two major works--Spirtes, Glymour, and Scheines 2000
(abbreviated SGS), and Pearl 2009--have been particularly
influential.



A causal model makes predictions about the behavior of a system. In
particular, a causal model entails the truth value, or the
probability, of counterfactual claims about the system; it predicts
the effects of interventions; and it entails the probabilistic
dependence or independence of variables included in the model. Causal
models also facilitate the inverse of these inferences: if we have
observed probabilistic correlations among variables, or the outcomes
of experimental interventions, we can determine which causal models
are consistent with these observations. The discussion will focus
on what it is possible to do in "in principle". For
example, we will consider the extent to which we can infer the correct
causal structure of a system, given perfect information about the
probability distribution over the variables in the system. This
ignores the very real problem of inferring the true probabilities from
finite sample data. In addition, the entry will discuss the application of
causal models to the logic of counterfactuals, the analysis of
causation, and decision theory.


## 2. Basic Tools



This section introduces some of the basic formal tools used in causal
modeling, as well as terminology and notational conventions.


### 2.1 Variables, Logic, and Language



*Variables* are the basic building blocks of causal models.
They will be represented by italicized upper case letters. A variable
is a function that can take a variety of values. The values of a
variable can represent the occurrence or non-occurrence of an event, a
range of incompatible events, a property of an individual or of a
population of individuals, or a quantitative value. For instance, we
might want to model a situation in which Suzy throws a stone and a
window breaks, and have variables *S* and *W* such that:


* \(S = 1\) represents Suzy throwing a rock; \(S = 0\) represents
her not throwing
* \(W = 1\) represents the window breaking; \(W = 0\) represents the
window remaining intact.



If we are modeling the influence of education on income in the United
States, we might use variables *E* and *I* such that:


* \(E(i) = 0\) if individual *i* has no high school education;
1 if an individual has completed high school; 2 if an individual has
had some college education; 3 if an individual has a bachelor's
degree; 4 if an individual has a master's degree; and 5 if an
individual as a doctorate (including the highest degrees in law and
medicine).
* \(I(i) = x\) if individual *i* has a pre-tax income of
$*x* per year.



The set of possible values of a variable is the *range* of that
variable. We will usually assume that variables have finitely many
possible values, as this will keep the mathematics and the exposition
simpler. However, causal models can also feature continuous variables,
and in some cases this makes an important difference.



A *world* is a complete specification of a causal model; the
details will depend upon the type of model. For now, we note that a
world will include, *inter alia*, an assignment of values to
all of the variables in the model. If the variables represent the
properties of individuals in a population, a world will include an
assignment of values to every variable, for every individual in the
population. A variable can then be understood as a function whose
domain is a set of worlds, or a set of worlds and individuals.



If *X* is a variable in a causal model, and *x* is a
particular value in the range of *X*, then \(X = x\) is an
*atomic proposition*. The logical operations of negation
("not"), conjunction ("and"), disjunction
("or"), the material conditional
("if...then..."), and the biconditional
("if and only if") are represented by
"\({\sim}\)", "&", "\(\lor\)",
"\(\supset\)", and "\(\equiv\)"
respectively. Any proposition built out of atomic propositions and
these logical operators will be called a *Boolean*
proposition. Note that when the variables are defined over individuals
in a population, reference to an individual is *not* included
in a proposition; rather, the proposition as a whole is true or false
of the various individuals in the population.



We will use basic notation from set theory. Sets will appear in
boldface.


* \(\mathbf{\varnothing}\) 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 \(\mathbf{\varnothing}\) 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.



If \(\bS = \{x\_1 , \ldots ,x\_n\}\) is a set of values in the range of
*X*, then \(X \in \bS\) is used as shorthand for the disjunction
of propositions of the form \(X = x\_i\), for \(i = 1,\ldots\), n.
Boldface represents *ordered sets* or *vectors*. If
\(\bX = \{X\_1 , \ldots ,X\_n\}\) is a vector of variables, and \(\bx =
\{x\_1 , \ldots ,x\_n\}\) is a vector of values, with each value \(x\_i\)
in the range of the corresponding variable \(X\_i\), then \(\bX = \bx\)
is the conjunction of propositions of the form \(X\_i = x\_i\).


### 2.2 Probability



In
 section 4,
 we will consider causal models that include probability. Probability
is a function, P, that assigns values between zero and one, inclusive.
The domain of a probability function is a set of propositions that
will include all of the Boolean propositions described above, but
perhaps others as well.



Some standard properties of probability are the following:


* If *A* is a contradiction, then \(\Pr(A) = 0\).
* If *A* is a tautology, then \(\Pr(A) = 1\).
* If \(\Pr(A \amp B) = 0\), then \(\Pr(A \lor B) = \Pr(A) +
\Pr(B)\).
* \(\Pr({\sim}A) = 1 - \Pr(A)\).



Some further definitions:


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

 

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


We will ignore problems that might arise when \(\Pr(B) = 0\).
* *A* and *B* are *probabilistically independent*
(with respect to \(\Pr\)) just in case \(\Pr(A \amp B) = \Pr(A) \times
\Pr(B). A\) and *B* are *probabilistically dependent* or
*correlated* otherwise. If \(\Pr(B) \gt 0\), then *A* and
*B* will be independent just in case \(\Pr(A \mid B) =
\Pr(A)\).
* Variables *X* and *Y* are probabilistically independent
just in case all propositions of the form \(X = x\) and \(Y = y\) are
probabilistically independent.
* *A* and *B* are *probabilistically independent
conditional on* *C* just in case \(\Pr(A \amp B \mid C) =
\Pr(A \mid C) \times \Pr(B \mid C)\). If \(\Pr(B \amp C) \gt 0\), this
is equivalent to \(\Pr(A \mid B \amp C) = \Pr(A \mid C)\). Following
the terminology of Reichenbach (1956), we will also say that
**C* screens off *B* from *A** when these
equalities hold. Conditional independence among variables is defined
analogously.



As a convenient shorthand, a probabilistic statement that contains
only a variable or set of variables, but no values, will be understood
as having 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

 

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


as shorthand for

 

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


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



We will not presuppose any particular interpretation of probability
(see the entry on
 interpretations of probability),
 but we will assume that frequencies in appropriately chosen samples
provide evidence about the underlying probabilities. For instance,
suppose there is a causal model including the variables *E*
and *I* described above, with \(\Pr(E = 3) = .25\). Then we expect
that if we survey a large, randomly chosen sample of American adults, we
will find that approximately a quarter of them have a Bachelor's
degree, but no higher degree. If the survey produces a sample frequency
that substantially differs from this, we have evidence that the model
is inaccurate.


### 2.3 Graphs



If \(\bV\) is the set of variables included in a causal model, one way
to represent the causal relationships among the variables in \(\bV\)
is by a *graph*. Although we will introduce and use graphs in
 section 3,
 they will play a more prominent role in
 section 4.
 We will discuss 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 1
 shows a directed graph on variable set \(\bV = \{S, T, W, X, Y,
Z\}\).



![a diagram where S has an arrow pointing north to T; T has an arrow pointing northwest to X and northeast to Y; Y has an arrow pointing northeast to Z; W has an arrow pointing north to Z and northwest to Y](fig1.svg)

Figure 1




A *path* in a directed graph is a non-repeating sequence of
arrows that have endpoints in common. For example, in
 Figure 1
 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 point in the same direction; 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. Such a
directed path is called a *cycle*. The graph in Figure 1
contains no cycles, and hence 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 from *X* to *Y*. \(\bPA(Y)\)
will denote the set of all parents of *Y*. In
 Figure 1,
 \(\bPA(Y) = \{T, W\}\). *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 a descendant
of itself. \(\bDE(X)\) denotes the set of all descendants of *X*.
In Figure 1, \(\bDE(T) = \{T,X, Y, Z\}\).



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: *Y* may be a direct cause of *Z*
relative to variable set \(\bV\), but not relative to variable set
\(\bV'\) that includes some additional variable(s) that mediate the
influence of *Y* on \(Z.\) As we develop our account of graphical
causal models in more detail, we will be able to say more precisely
what it means for one variable to be a direct cause of another. While
we will not define "cause", causal models presuppose a
broadly *difference-making* notion of causation, rather than a
causal process notion (Salmon 1984, Dowe 2000) or a mechanistic notion
(Machamer, Darden, & Craver 2000; Glennan 2017). We will call the
system of direct causal relations represented in a DAG such as
 Figure 1
 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 2(a)).
 An ADMG on the variable set \(\bV = \{X, Y\}\) will look like
 Figure 2(b).





![diagram with L having an arrow pointing northwest to X and northeast to Y](fig2a.svg)

(a)




![X is to the right of Y and a double-headed curved arrow connects the two](fig2b.svg)

(b)





Figure 2




We can be a bit more precise. 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*.



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 2(b).
 *Directed path* retains the same meaning, and a directed path
cannot contain double-headed arrows.



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 1
 represents that *W* is a direct cause of *Y*, that *X*
is not a direct cause of *Y*, and that *there are no latent
common causes*. The absence of double-headed arrows from Figure 1
does not show merely that we have chosen not to include latent common
causes in our representation; it shows that there are no latent common
causes.


## 3. Deterministic Structural Equation Models



In this section, we introduce deterministic *structural equation
models* (SEMs), postponing discussion of probability until
 Section 4.
 We will consider two applications of deterministic SEMs: the logic of
counterfactuals, and the analysis of actual causation.


### 3.1 Introduction to SEMs



A SEM characterizes a causal system with a set of variables, and a set
of equations describing how each variable depends upon its immediate
causal predecessors. Consider a gas grill, used to cook meat. We can
describe the operations of the grill using the following
variables:


* *Gas connected* (1 if yes, 0 if no)
* *Gas knob* (0 for off, 1 for low, 2 for medium, 3 for
high)
* *Gas level* (0 for off, 1 for low, 2 for medium, 3 for
high)
* *Igniter* (1 if pressed, 0 if not)
* *Flame* (0 for off, 1 for low, 2 for medium, 3 for
high)
* *Meat on* (0 for no, 1 for yes)
* *Meat cooked* (0 for raw, 1 for rare, 2 for medium, 3 for
well done)



Thus, for example, *Gas knob* = 1 means that the gas knob is
set to low; *Igniter* = 1 means that the igniter is pressed,
and so on. Then the equations might be:


* *Gas level* = *Gas connected* \(\times\) *Gas
knob*
* *Flame* = *Gas level* \(\times\)
*Igniter*
* *Meat cooked* = *Flame* \(\times\) *Meat
on*



The last equation, for example, tells us that if the meat is not put
on the grill, it will remain raw (*Meat cooked* = 0). If the
meat is put on the grill, then it will get cooked according to the
level of the flame: if the flame is low (*Flame* = 1), the meat
will be rare (*Meat cooked* = 1), and so on.



By convention each equation has one effect variable on the left hand
side, and one or more cause variables on the right hand side. We also
exclude from each equation any variable that makes no difference to
the value of the effect variable. For example, the equation for
*Gas level* could be written as *Gas level* = (*Gas
connected* \(\times\) *Gas knob*) \(+\) (0 \(\times\) *Meat
cooked*); but since the value of *Meat cooked* makes no
difference to the value of *Gas level* in this equation, we
omit the variable *Meat cooked*. A SEM is *acyclic* if
the variables can be ordered so that variables never appear on the
left hand side of an equation after they have appeared on the right.
Our example is acyclic, as shown by the ordering of variables given
above. In what follows, we will assume that SEMs are acyclic, unless
stated otherwise.



We can represent this system of equations as a DAG
 (Figure 3):



![diagram: 'Gas connected' has an arrow pointing northeast to 'Gas level'; 'Gas knob' has an arrow pointing northwest to the same 'Gas level'; 'Gas level' has an arrow pointing northeast to 'Flame'; 'Igniter' has an arrow pointing northwest to the same 'Flame'; 'Flame' has an arrow pointing northeast to 'Meat cooked'; 'Meat on' has an arrow pointing northwest to the same 'Meat cooked'](fig3.svg)

Figure 3




An arrow is drawn from variable *X* to variable *Y*
just in case *X* figures as an argument in the equation for
*Y*. The graph contains strictly less information than the set of
equations; in particular, the DAG gives us qualitative information
about which variables depend upon which others, but it does not tell
us anything about the functional form of the dependence.



The variables in a model will typically depend upon further variables
that are not explicitly included in the model. For instance, the level
of the flame will also depend upon the presence of oxygen. Variables that
are not explicitly represented in the model are assumed to be fixed at
values that make the equations appropriate. For example, in our model
of the gas grill, oxygen is assumed to be present in sufficient
quantity to sustain a flame ranging in intensity from low to high.



In our example, the variables *Gas level*, *Flame,* and
*Meat cooked* are *endogenous*, meaning that their
values are determined by other variables in the model. *Gas
connected, Gas knob, Igniter,* and *Meat on* are
*exogenous*, meaning that their values are determined outside
of the system. In all of the models that we will consider in
 section 3,
 the values of the exogenous variables are given or otherwise
known.



Following Halpern (2016), we will call an assignment of values to the
exogenous variables a *context*. In an acyclic SEM, a context
uniquely determines the values of all the variables in the model. An
acyclic SEM together with a context is a *world* (what Halpern
2016 calls a "causal setting"). For instance, if we add
the setting


* *Gas connected* = 1
* *Gas knob* = 3
* *Igniter* = 1
* *Meat on* = 1



to our three equations above, we get a world in which *Gas
level* = 3, *Flame* = 3, and *Meat cooked* = 3.



The distinctively causal or "structural" content of a SEM
derives from the way in which *interventions* are represented.
To intervene on a variable is to set the value of that variable by a
process that overrides the usual causal structure, without interfering
with the causal processes governing the other variables. More
precisely, an intervention on a variable *X* overrides the normal
equation for *X*, while leaving the other equations unchanged.
For example, to intervene on the variable *Flame* in our
example would be to set the flame to a specified level regardless of
whether the igniter is pressed or what the gas level is. (Perhaps, for
example, one could pour kerosene into the grill and light it with a
match.) Woodward (2003) proposes that we think of an intervention as a
causal process that operates independently of the other variables in
the model. Randomized controlled trials aim to intervene in this
sense. For example, a randomized controlled trial to test the efficacy
of a drug for hypertension aims to determine whether each subject
takes the drug (rather than a placebo) by a random process such as a
coin flip. Factors such as education and health insurance that
normally influence whether someone takes the drug no longer play this
role for subjects in the trial population. Alternately, we could
follow the approach of Lewis (1979) and think of an intervention
setting the value of a variable by a minor "miracle".



To represent an intervention on a variable, we *replace* the
equation for that variable with a new equation stating the value to
which the variable is set. For example, if we intervene to set the
level of flame at *low*, we would represent this by replacing
the equation *Flame* = *Gas level* \(\times\)
*Igniter* with *Flame* = 1. This creates a new causal
structure in which *Flame* is an exogenous variable;
graphically, we can think of the intervention as "breaking the
arrows" pointing into *Flame*. The new system of
equations can then be solved to discover what values the other
variables would take as a result of the intervention. In the world
described above, our intervention would produce the following set of
equations:


* *Gas connected* = 1
* *Gas knob* = 3
* *Igniter* = 1
* *Meat on* = 1
* *Gas level* = *Gas connected* x *Gas
knob*
* *Flame* = *Gas level* x *Igniter*
* *Flame* = 1
* *Meat cooked* = *Flame* x *Meat
on*



We have struck through the original equation for *Flame* to
show that it is no longer operative. The result is a new world with a
modified causal structure, with *Gas level* = 3, *Flame*
= 1, and *Meat cooked* = 1. Since the equation connecting
*Flame* to its causes is removed, any changes introduced by
setting *Flame* to 1 will only propagate forward through the
model to the descendants of *Flame*. The intervention changes
the values of *Flame* and *Meat cooked*, but it does not
affect the values of the other variables. We can represent
interventions on multiple variables in the same way, replacing the
equations for all of the variables intervened on.



Interventions help to give content to the arrows in the corresponding
DAG. If variable \(X\_i\) is a parent of \(X\_j\), this means that there
exists some setting for all of the other variables in the model, such
that when we set those variables to those values by means of an
intervention, intervening on \(X\_i\) can still make a difference for
the value of \(X\_j\). For example, in our original model, *Gas
level* is a parent of *Flame*. If we set the value of
*Igniter* to 1 by means of an intervention, and set *Gas
knob, Gas connected, Meat on,* and *Meat cooked* to any
values at all, then intervening on the value of *Gas level*
makes a difference for the value of *Flame*. Setting the value
of *Gas level* to 1 would yield a value of 1 for
*Flame*; setting *Gas level* to 2 yields a
*Flame* of 2; and so on.


### 3.2 Structural Counterfactuals



A counterfactual is a proposition in the form of a subjunctive
conditional. The antecedent posits some circumstance, typically one
that is contrary to fact. For example, in our gas grill world, the
flame was high, and the meat was well done. We might reason: "if
the flame had been set to low, the meat would have been rare".
The antecedent posits a hypothetical state of affairs, and the
consequent describes what would have happened in that hypothetical
situation.



Deterministic SEMs naturally give rise to a logic of counterfactuals.
These counterfactuals are called *structural* counterfactuals
or *interventionist* counterfactuals. Structural
counterfactuals are similar in some ways to what Lewis (1979) calls
*non-backtracking* counterfactuals. In a non-backtracking
counterfactual, one does not reason backwards from a counterfactual
supposition to draw conclusions about the causes of the hypothetical
situation. For instance, one would not reason "If the meat had
been cooked rare, then the flame would have been set to low".
Lewis (1979) proposes that we think of the antecedent of a
counterfactual as coming about through a minor "miracle".
The formalism for representing interventions described in the previous
section prevents backtracking from effects to causes.



The logic of structural counterfactuals has been developed by Galles
and Pearl (1998), Halpern (2000), Briggs (2012), and Zhang
(2013a). This section will focus on Briggs' formulation; it has
the richest language, but unlike the other approaches it can not be
applied to causal models with cycles. Despite a shared concern with
non-backtracking counterfactuals, Briggs' logic differs in a
number of ways from the more familiar logic of counterfactuals
developed by Stalnaker (1968) and Lewis (1973b).



We interpret the counterfactual conditional \(A \boxright B\) as saying
that *B* would be true, if *A* were made true by an
intervention. The language of structural counterfactuals does not
allow the connective '\(A \boxright B\)' to appear in the antecedents of
counterfactuals. More precisely, we define well-formed formulas
(*wff*s) for the language inductively:


* Boolean propositions are *wff*s
* If *A* is a Boolean proposition, and *B* is a
*wff*, then \(A \boxright B\) is a *wff*



This means, for example, that \(A \boxright (B\boxright (C\boxright D))\)
is a *wff*, but \(A\boxright ((B\boxright C)\boxright D)\) is not,
since the embedded counterfactual in the consequent does not have a
Boolean proposition as an antecedent.



Consider the world of the gas grill, described in the previous
section:


* *Gas connected* = 1
* *Gas knob* = 3
* *Igniter* = 1
* *Meat on* = 1
* *Gas level* = *Gas connected* \(\times\) *Gas
knob*
* *Flame* = *Gas level* \(\times\)
*Igniter*
* *Meat cooked* = *Flame* \(\times\) *Meat
on*



To evaluate the counterfactual \({\textit{Flame} = 1} \boxright
{\textit{Meat cooked} = 1}\) (if the flame had been set to
*low*, the meat would have been cooked rare), we replace the
equation for *Flame* with the assignment *Flame* = 1. We
can then compute that *Meat cooked* = 1; the counterfactual is
true. If the antecedent is a conjunction of atomic propositions, such
as *Flame* = 1 and *Igniter* = 0, we replace all of the
relevant equations. A special case occurs when the antecedent conjoins
atomic propositions that assign different values to the same variable,
such as *Flame* = 1 and *Flame* = 2. In this case, the
antecedent is a contradiction, and the counterfactual is considered
trivially true.



If the antecedent is a disjunction of atomic propositions, or a
disjunction of conjunctions of atomic propositions, then the
consequent must be true when *every* possible intervention or
set of interventions described by the antecedent is performed.
Consider, for instance, 

 

\[
\begin{aligned}
(({\textit{Flame}= 1} \amp {\textit{Gas level}= 0}) \lor ({\textit{Flame}= 2} \amp {\textit{Meat on}= 0}))\\
{} \boxright
 ({\textit{Meat cooked}= 1} \lor {\textit{Meat cooked}= 2}).
\end{aligned}
\]


If we perform the first intervention, we compute that *Meat
cooked* = 1, so the consequent is true. However, if we perform the
second intervention, we compute that *Meat cooked* = 0. Hence
the counterfactual comes out false. Some negations are treated as
disjunctions for this purpose. For example, \({\sim}(\textit{Flame} =
1)\) would be treated in the same way as the disjunction



 \[{\textit{Flame} = 0} \lor {\textit{Flame} = 2} \lor {\textit{Flame} = 3}.\]


If the consequent contains a counterfactual, we iterate the procedure.
Consider the counterfactual: 

 

\[
{\textit{Flame} = 1} \boxright ({\textit{Gas level} = 0} \boxright ({\textit{Flame} = 2} \boxright {\textit{Meat cooked} = 1})).
\]


To evaluate this counterfactual, we first change the equation for
*Flame* to *Flame* = 1. Then we change the equation for
*Gas level* to *Gas level* = 0. Then we change the
equation for *Flame* again, from *Flame* = 1, to
*Flame* = 2. Finally, we compute that *Meat cooked* = 2,
so the counterfactual comes out false. Unlike the case where
*Flame* = 1 and *Flame* = 2 are conjoined in the
antecedent, the two different assignments for *Flame* do not
generate an impossible antecedent. In this case, the interventions are
performed in a specified order: *Flame* is first set to 1, and
*then* set to 2.



The differences between structural counterfactuals and Stalnaker-Lewis
counterfactuals stem from the following two features of structural
counterfactuals:


1. The antecedent of a counterfactual is always thought of as being
realized by an intervention, even if the antecedent is already true in
a given world. For instance, in our gas grill world, *Flame* =
3. Nonetheless, if we evaluate a counterfactual with antecedent
*Flame* = 3 in this world, we replace the equation for
*Flame* with *Flame* = 3.
2. The truth values of counterfactuals are determined solely by the
causal structures of worlds, together with the interventions specified
in the their antecedents. No further considerations of
*similarity* play a role. For example, the counterfactual 



\[ 
{\textit{Flame}= 1} \lor {\textit{Flame} = 2} \boxright {\textit{Flame}= 2} 
\]


would be false in our gas grill world (and indeed in all possible
worlds). We do not reason that a world in which *Flame* = 2 is
*closer* to our world (in which *Flame* = 3) than a
world in which *Flame* = 1.



These features of structural counterfactuals lead to some unusual
properties in the full language developed by Briggs (2012):


1. The analog of *modus ponens* fails for the structural
conditional; i.e., from *A* and \(A\boxright B\) we cannot infer
*B*. For example, in our gas grill world, *Flame* = 3 and

\[
 {\textit{Flame} = 3} \boxright {({\textit{Gas level} = 2} \boxright {\textit{Meat cooked} = 3})} 
 \]

 are both true, but 

\[
{\textit{Gas level} = 2} \boxright {\textit{Meat cooked} = 3} 
\]

 is false. To evaluate the
last counterfactual, we replace the equation for *Gas level*
with *Gas level* = 2. This results in *Flame* = 2 and
*Meat cooked* = 2. To evaluate the prior counterfactual, we
first substitute in the equation *Flame* = 3. Now, the value of
*Flame* no longer depends upon the value of *Gas level*,
so when we substitute in *Gas level* = 2, we get *Meat
cooked* = 3. Even though the actual value of *Flame* is 3,
we evaluate the counterfactual by intervening on *Flame* to set
it to 3. With this intervention in place, a further intervention on
*Gas level* makes no difference to *Flame* or *Meat
cooked.*
2. The substitution of logically equivalent propositions into the
antecedent of a counterfactual does not always preserve truth value.
For example, 

\[
{\textit{Gas level} = 2} \boxright {\textit{Meat cooked} = 2} 
\]

 is true, but 

\[\begin{align}
& ({\textit{Gas level} = 2} \amp ({\textit{Flame} = 2} \lor {{\sim}(\textit{Flame} =2))}\\
& \qquad \boxright\, {\textit{Meat cooked} = 2} 
\end{align}\]

 is false. The
first counterfactual does not require us to intervene on the value of
*Flame*, but the second counterfactual requires us to consider
interventions that set *Flame* to all of its possible
values.



To handle the second kind of case, Briggs (2012) defines a relation of
*exact* *equivalence* among Boolean propositions using
the state space semantics of Fine (2012). Within a world, the state
that makes a proposition true is the collection of values of variables
that contribute to the truth of the proposition. In our example world,
the state that makes *Gas level* = 3 true is the valuation
*Gas level* = 3. By contrast, the state that makes 



\[\textit{Gas level} = 3 \amp (\textit{Flame} = 2 \lor {\sim}(\textit{Flame} = 2))\] 


true includes both *Gas level* = 3 and *Flame* = 3.
Propositions are exactly equivalent if they are made true by the same
states in all possible worlds. The truth value of a counterfactual is
preserved when exactly equivalent propositions are substituted into
the antecedent.



Briggs (2012) provides a sound and complete axiomatization for
structural counterfactuals in acyclic SEMs. The axioms and inference
rules of this system are presented in
 Supplement on Briggs' Axiomatization.


### 3.3 Actual Causation



Many philosophers and legal theorists have been interested in the
relation of *actual causation*. This concerns the assignment of
causal responsibility for some event that occurs, based on how events
actually play out. For example, suppose that Billy and Suzy are both
holding rocks. Suzy throws her rock at a window, but Billy does not.
Suzy's rock hits the window, which breaks. Then Suzy's
throw was the actual cause of the window breaking.



We can represent this story easily enough with a structural equation
model. For variables, we choose:


* \(B = 1\) if Billy throws, 0 if he doesn't
* \(S = 1\) if Suzy throws, 0 if she doesn't
* \(W = 1\) if the window shatters, 0 if it doesn't



Our context and equation will then be:


* \(B = 0\)
* \(S = 1\)
* \(W = \max(B, S)\)



The equation for *W* tells us that the window would shatter if
either Billy or Suzy throws their rock. The corresponding DAG is shown
in
 Figure 4




![diagram: B has an arrow pointing northeast to W and S has an arrow pointing northwest to the same W](fig4.svg)

Figure 4




But we cannot simply read off the the relation of actual causation
from the graph or from the equations. For example, the arrow from
*B* to *W* in
 Figure 4
 cannot be interpreted as saying that Billy's (in)action is an
actual cause of the window breaking. Note that while it is common to
distinguish between singular or token causation, and general or
type-level causation (see, e.g., Eells 1991, Introduction), that is
not what is at issue here. Our causal model does not represent any
kind of causal generalization: it represents the actual and possible
actions of Billy and Suzy at one particular place and time. Actual
causation is not just causal structure of the single case. A further
criterion for actual causation, defined in terms of the causal
structure together with the actual values of the variables, is
needed.



Following Lewis (1973a), it is natural to try to analyze the relation
of actual causation in terms of *counterfactual dependence.* In
our model, the following propositions are all true:


* \(S = 1\)
* \(W = 1\)
* \({S = 0}\boxright {W = 0}\)



In words: Suzy threw her rock, the window shattered, and if Suzy
hadn't thrown her rock, the window wouldn't have
shattered. In general, we might attempt to analyze actual causation as
follows:




\(X = x\) is an actual cause of \(Y = y\) in world *w* just in
case:


* *X* and *Y* are different variables
* \(X = x\) and \(Y = y\) in *w*
* There exist \(x' \ne x\) and \(y' \ne y\) such that \({X = x'}
\boxright {Y = y'}\) is true in *w*




Unfortunately, this simple analysis will not work, for familiar
reasons involving *preemption* and *overdetermination*.
Here is an illustration of each:




*Preemption:* Billy decides that he will give Suzy the
opportunity to throw first. If Suzy throws her rock, he will not
throw, but if she doesn't throw, he will throw and his rock will
shatter the window. In fact, Suzy throws her rock, which shatters the
window. Billy does not throw.



*Overdetermination:* Billy and Suzy throw their rocks
simultaneously. Their rocks hit the window at the same time,
shattering it. Either rock by itself would have been sufficient to
shatter the window.




In both of these cases, Suzy's throw is an actual cause
of the window's shattering, but the shattering does not
counterfactually depend upon her throw: if Suzy hadn't thrown
her rock, Billy's rock would have shattered the window. Much of
the work on counterfactual theories of causation since 1973 has been
devoted to addressing these problems.



A number of authors have used SEMs to try to formulate adequate
analyses of actual causation in terms of counterfactuals, including
Beckers & Vennekens (2018), Glymour & Wimberly (2007), Halpern
(2016), Halpern & Pearl (2001, 2005), Hitchcock (2001), Pearl
(2009: Chapter 10), Weslake (forthcoming), and Woodward (2003: Chapter
2). As an illustration, consider one analysis based closely on
a proposal presented in Halpern (2016):




(**AC**) \(X = x\) is an actual cause of \(Y = y\) in
world *w* just in case:



1. *X* and *Y* are different variables
2. \(X = x\) and \(Y = y\) in *w*
3. There exist disjoint sets of variables \(\bX\) and \(\bZ\) with
\(X \in \bX\), with values \(\bX = \bx\) and \(\bZ = \bz\) in
*w*, such that:
	1. There exists \(\bx' \ne \bx\) such that 
	
	\[({\bX = \bx'} \amp {\bZ = \bz}) \boxright {Y \ne y}\] 
	
	 is true in
	*w*
	2. No subset of \(\bX\) satisfies (3a)





That is, *X* belongs to a minimal set of variables \(\bX\), such
that when we intervene to hold the variables in \(\bZ\) fixed at the
values they actually take in *w*, *Y* counterfactually
depends upon the values of the variables in \(\bX.\) We will
illustrate this account with our examples of preemption and
overdetermination.



In *Preemption*, let the variables *B*, *S*, and
*W* be defined as above. Our context and equations are:


* \(S = 1\)
* \(B = 1 - S\)
* \(W = \max(B, S)\)



That is: Suzy throws her rock; Billy will throw his rock if Suzy
doesn't; and the window will shatter if either throws their
rock. The DAG is shown in
 Figure 5.



![diagram: B has an arrow pointing northeast to W and S has an arrow pointing northwest to the same W. S also has an arrow pointing west to the same B.](fig5.svg)

Figure 5




We want to show that \(S = 1\) is an actual cause of \(W = 1\).
Conditions AC(1) and AC(2) are clearly satisfied. For condition AC(3),
we choose \(\bX = \{S\}\) and \(\bZ = \{B\}\). Since \(B = 0\) in
*Preemption*, we want to fix \(B = 0\) while varying *S*.
We can see easily that \({S = 0} \amp {B = 0} \boxright {W = 0}\):
replacing the two equations for *B* and *S* with \(B = 0\)
and \(S = 0\), the solution yields \(W = 0\). In words, this
counterfactual says that if neither Billy nor Suzy had thrown their
rock, the window would not have shattered. Thus condition AC(3a) is
satisfied. AC(3b) is satisfied trivially, since \(\bX = \{S\}\) is a
singleton set.



Here is how AC works in this example. *S* influences *W*
along two different paths: the direct path \(S \rightarrow W\) and the
indirect path \(S \rightarrow B \rightarrow W\). These two paths
interact in such a way that they cancel each other out, and the value
of *S* makes no net difference to the value of *W*. However,
by holding *B* fixed at its actual value of 0, we eliminate the
influence of *S* on *W* along that path. The result is that
we isolate the contribution that *S* made to *W* along the
direct path. AC defines actual causation as a particular kind of
*path-specific* effect.



To treat *Overdetermination*, let *B*, *S*, and
*W* keep the same meanings. Our setting and equation will be:


* \(B = 1\)
* \(S = 1\)
* \(W = \max(B, S)\)



The graph is the same as that shown in
 Figure 4
 above. Again, we want to show that \(S = 1\) is an actual cause of
\(W = 1\). Conditions AC(1) and AC(2) are obviously satisfied. For
AC(3), we choose \(\bX = \{B, S\}\) and \(\bZ = \varnothing\). For
condition AC(3a), we choose for our alternate setting \(\bX = \bx'\) \(B =
0\) and \(S = 0\). Once again, the counterfactual \({S = 0} \amp {B =
0} \boxright {W = 0}\) is true. Now, for AC(3b) we must show that \(\bX
= \{B, S\}\) is minimal. It is easy to check that \(\{B\}\) alone
won't satisfy AC(3a). Whether we take \(\bZ = \varnothing\) or
\(\bZ = \{S\}\), changing *B* to 0 (perhaps while also setting
*S* to 1) will not change the value of *W*. A parallel
argument shows that \(\{S\}\) alone won't satisfy AC(3a) either.
The key idea here is that *S* is a member of a minimal set of
variables that need to be changed in order to change the value of
*W*.



Despite these successes, none of the analyses of actual causation
developed so far perfectly captures our pre-theoretic intuitions in
every case. One strategy that has been pursued by a number of authors
is to incorporate some distinction between *default* and
*deviant* values of variables, or between *normal* and
*abnormal* conditions. See, e.g., Hall (2007), Halpern (2008;
2016: Chapter 3), Halpern & Hitchcock (2015), Hitchcock (2007),
and Menzies (2004). Blanchard & Schaffer (2017) present arguments
against this approach. Glymour et al. (2010) raise a number of
problems for the project of trying to analyze actual causation.


## 4. Probabilistic Causal Models



In this section, we will discuss causal models that incorporate
probability in some way. Probability may be used to represent our
uncertainty about the value of unobserved variables in a particular
case, or the distribution of variable values in a population. Often we
are interested in when some feature of the causal structure of a
system can be *identified* from the probability distribution
over values of variables, perhaps in conjunction with background
assumptions and other observations. For example, we may know the
probability distribution over a set of variables \(\bV\), and want to
know which causal structures over the variables in \(\bV\) are
compatible with the distribution. 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. We will not address these important
practical concerns here. Rather, our focus will be on what it is
possible to infer from probabilities, in principle if not in practice.
We will also consider the application of probabilistic causal models
to decision theory and counterfactuals.


### 4.1 Structural Equation Models with Random Errors



We can introduce probability into a SEM by means of a probability
distribution over the exogenous variables.



Let \(\bV = \{X\_1, X\_2 ,\ldots ,X\_n\}\) be a set of endogenous
variables, and \(\bU = \{U\_1, U\_2 ,\ldots ,U\_n\}\) a corresponding set
of exogenous variables. Suppose that each endogenous variable \(X\_i\)
is a function of its parents in \(\bV\) together with \(U\_i\), that
is:

 

\[X\_i = f\_i (\bPA(X\_i), U\_i).\]


As a general rule, our graphical representation of this SEM will
include only the endogenous variables \(\bV\), and we use
\(\bPA(X\_i)\) to denote the set of *endogenous* parents of
\(X\_i . U\_i\) is sometimes called an *error* variable for
\(X\_i\): it is responsible for any difference between the actual value
of \(X\_i\) and the value predicted on the basis of \(\bPA(X\_i)\)
alone. We may think of \(U\_i\) as encapsulating all of the causes of
\(X\_i\) that are not included in \(\bV\). The assumption that each
endogenous variable has exactly one error variable is innocuous. If
necessary, \(U\_i\) can be a vector of variables. For example, if
\(Y\_1\), \(Y\_2\), and \(Y\_3\) are all causes of \(X\_i\) that are not
included in \(\bV\), we can let \(U\_i = \langle Y\_1, Y\_2,
Y\_3\rangle\). Moreover, the error variables need not be distinct or
independent from one another.



Assuming that the system of equations is acyclic, an assignment of
values to the exogenous variables \(U\_1\), \(U\_2\),... ,\(U\_n\)
uniquely determines the values of all the variables in the model.
Then, if we have a probability distribution \(\Pr'\) over the values
of variables in \(\bU\), this will induce a unique probability
distribution P on \(\bV\).


### 4.2 The Markov Condition



Suppose we have a SEM with endogenous variables \(\bV\), exogenous
variables \(\bU\), probability distribution P on \(\bU\) and \(\bV\)
as described in the previous section, and DAG \(\bG\) representing the
causal structure on \(\bV\). Pearl and Verma (1991) prove that if the
error variables \(U\_i\) are probabilistically independent in P, then
the probability distribution on \(\bV\) will satisfy the *Markov
Condition* (MC) with respect to \(\bG\). The Markov Condition has
several formulations, which are equivalent when \(\bG\) is a a DAG
(Pearl 1988):




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



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, \(\Pr(X\_i
\mid \bPA(X\_i))\), we can compute the complete joint distribution over
all of the variables. It is relatively easy to see that MCFactorization follows from MCScreening\_off. Since \(\bG\)
is acyclic, we may re-label the subscripts on the variables so that
they are ordered from 'earlier' to 'later',
with only earlier variables being ancestors of later ones. It follows
from the probability calculus that 

\[\Pr(X\_1, X\_2 , \ldots ,X\_n) = \Pr(X\_1) \times \Pr(X\_2 \mid X\_1) \times \ldots \times \Pr(X\_n \mid X\_1, X\_2 , \ldots ,X\_{n-1})\] 

 (this is a version of
the theorem of total probability). For each term \(\Pr(X\_i \mid X\_1,
X\_2 , \ldots ,X\_{i-1})\), our ordering ensures that all of the parents
of \(X\_i\) will be included on the right hand side, and none of its
descendants will. MCScreening\_off then tells us that we can
eliminate all of the terms from the right hand side except for the
parents of \(X\_i\).



MC*d*-separation introduces the graphical notion of
*d*-separation. 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 \(X\_i\). In other
words, \(X\_i\) is a collider just in case the arrows converge on
\(X\_i\) in the path. Let \(\bX, \bY\), and \(\bZ\) be disjoint subsets
of \(\bV\). \(\bZ\) *d*-separates \(\bX\) and \(\bY\) just in case every
path \(\langle X\_1 , \ldots ,X\_k\rangle\) from a variable in \(\bX\)
to a variable in \(\bY\) 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 \(\bZ\). Any path that meets this
condition is said to be *blocked* by \(\bZ\). If \(\bZ\) does
not *d*-separate \(\bX\) and \(\bY\), then \(\bX\) and \(\bY\)
are *d-connected* by \(\bZ\).



Note that MC provides sufficient conditions for variables to be
probabilistically independent, conditional on others, but no necessary
condition.



Here are some illustrations:



![diagram: T has an arrow pointing north to W; W has a long arrow pointing northeast to Z; W also has an arrow pointing northwest to X; X has an arrow pointing northwest to Y](fig6.svg)

Figure 6




In
 Figure 6,
 MC implies that *X* screens *Y* off from all of the other
variables, and *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.
*T* does not necessarily screen *Y* off from *Z* (or
indeed anything from anything).



![diagram: X has an arrow pointing northeast to Y and Z has an arrow pointing northwest to the same Y.](fig7.svg)

Figure 7




In
 Figure 7,
 MC entails that *X* and *Z* will be unconditionally
independent, but not that they will be independent conditional on
*Y*. This is most easily seen from MC*d*-separation.



Let \(V\_i\) and \(V\_j\) be two distinct variables in \(\bV\), with
corresponding exogenous error variables \(U\_i\) and \(U\_j\),
representing causes of \(V\_i\) and \(V\_j\) that are excluded from the
\(\bV\). Suppose \(V\_i\) and \(V\_j\) share at least one common cause
that is excluded from \(\bV\). In this case, we would not expect
\(U\_i\) and \(U\_j\) to be probabilistically independent, and the
theorem of Pearl and Verma (1991) would not apply. In this case, the
causal relationship among the variables in \(\bV\) would not be
appropriately represented by a DAG, but would require an acyclic
directed mixed graph (ADMG) with a double-headed arrow connecting
\(V\_i\) and \(V\_j\). We will discuss this kind of case in more detail
in
 Section 4.6
 below.



MC is not expected to hold for arbitrary sets of variables \(\bV\),
even when the DAG \(\bG\) 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-Podolski-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*.



Both SGS (2000) and Pearl (2009) contain statements of a principle
called the *Causal Markov Condition* (CMC). The statements are
in fact quite different from one another. In Pearl's
formulation, (CMC) is just a statement of the mathematical theorem
described above: 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 \(\bG\) representing the functional dependence relations among
the variables in \(\bV\). 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 above; \(\bG\)
is a DAG representing the causal structure on \(\bV\); and P is the
empirical probability distribution resulting from this causal
structure; then P can be expected to satisfy MC relative to \(\bG\).
They defend this assumption in (at least) two 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.*


### 4.3 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
conditions that give necessary conditions for probabilistic
independence. (This is employing the terminology of Spirtes et
al. (SGS 2000). Pearl (2009) contains a "Minimality
Condition" that is slightly different from the one described
here.)



(i) *The Minimality Condition*. Suppose that the DAG \(\bG\) 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. 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. 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 \(\bZ
= \bPA(Y) \setminus \{X\}\), there exist \(\bz\), *y*, *x*,
\(x'\) such that \(\Pr(Y = y \mid X = x \amp \bZ = \bz) \ne \Pr(Y = y
\mid X = x' \amp \bZ = \bz)\).



(ii) *The Faithfulness Condition*. The Faithfulness Condition
(FC) is the converse of the Markov Condition: it 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 *Z* are unconditionally independent of one another,
but dependent, conditional upon *Y*. (The other two variable
pairs are dependent, both conditionally and unconditionally.) The
graph shown in
 Figure 8
 does not satisfy FC 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
*Z*. This can be seen by noting that *X* and *Z* are
*d*-connected (by the empty set): neither the path \(X
\rightarrow Z\) nor \(X \rightarrow Y\rightarrow Z\) is blocked (by
the empty set). By contrast, the graph shown in
 Figure 7
 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, FC is strictly stronger than the Minimality
Condition.



![diagram: X has an arrow pointing northeast to Y and another arrow pointing east to Z; Y has an arrow pointing southeast to Z.](fig8.svg)

Figure 8




Here are some other examples: In
 Figure 6
 above, there is a path \(W\rightarrow X\rightarrow Y\); FC implies
that *W* and *Y* should be probabilistically dependent. In
 Figure 7,
 FC implies that *X* and *Z* should be dependent,
conditional on *Y*.



FC can fail if the probabilistic parameters in a causal model are just
so. In
 Figure 8,
 for example, *X* influences *Z* along two different
directed paths. If the effect of one path is to exactly undo the
influence along the other path, then *X* and *Z* will be
probabilistically independent. If the underlying SEM is linear,
Spirtes et al. (SGS 2000: Theorem 3.2) prove that the set of
parameters for which Faithfulness is violated has Lebesgue measure 0.
Nonetheless, parameter values leading to violations of FC are
*possible*, so FC does not seem plausible as a metaphysical or
conceptual constraint upon the connection between causation and
probabilities. It is, rather, a *methodological* principle:
Given a distribution on \(\{X, Y, Z\}\) in which *X* and *Z*
are independent, we should prefer the causal structure depicted in
 Figure 7
 to the one in 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 *Z* rather than one that is merely
*consistent* with independence. See Zhang and Spirtes 2016 for
comprehensive discussion of the role of FC.



Violations of FC are often detectable in principle. For example,
suppose that the true causal structure is that shown in
 Figure 7,
 and that the probability distribution over *X*, *Y*, and
*Z* exhibits all of the conditional independence relations
required by MC. Suppose, moreover, that *X* and *Z* are
independent, conditional upon *Y*. This conditional independence
relation is not entailed by MC, so it constitutes a violation of FC.
It turns out that there is no DAG that is faithful to this probability
distribution. This tips us off that there is a violation of FC. While
we will not be able to infer the correct causal structure, we will at
least avoid inferring an incorrect one in this case. For details, see
Steel 2006, Zhang & Spirtes 2008, and Zhang 2013b.



Researchers have explored the consequences of adopting a variety of
assumptions that are weaker than FC; see for example Ramsey et al.
2006, Spirtes & Zhang 2014, and Zhalama et al. 2016.


### 4.4 Identifiability of Causal Structure



If we have a set of variables \(\bV\) and know the probability
distribution P on \(\bV\), what can we infer about the causal
structure on \(\bV\)? This epistemological question is closely related
to the metaphysical question of whether it is possible to
*reduce* causation to probability (as, e.g., Reichenbach 1956
and Suppes 1970 proposed).



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 variables, 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.




It is relatively easy to see why this holds. For each variable
\(X\_i\), its parents must come from among the variables with lower
time indices, call them \(X\_1 ,\ldots ,X\_{i-1}\). Any variables in
this group that are not parents of \(X\_i\) will be nondescendants of
\(X\_i\); hence they will be screened off from \(X\_i\) by its parents
(from MCScreening\_off). Thus we can start with the
distributions \(\Pr(X\_i\mid X\_1 ,\ldots ,X\_{i-1})\), and then weed out
any variables from the right hand side that make no difference to the
probability distribution over \(X\_i\). By the Minimality Condition, we
know that the variables so weeded are not parents of \(X\_i\). Those
variables that remain are the parents of \(X\_i\) in \(\bG\).



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. In other words, it will be the set of all DAGs \(\bG\) such that P
satisfies MC and FC with respect to \(\bG\). The PC algorithm
described by SGS (2000: 84-85) is one algorithm that generates
the Markov equivalence class for any probability distribution with a
non-empty Markov equivalence class.



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:

 

\[
X \rightarrow Y \rightarrow Z\\
X \leftarrow Y \leftarrow Z\\
X \leftarrow Y \rightarrow Z
\]


We cannot determine from the probability distribution, together with
MC and FC, 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:

 

\[
X \rightarrow Y \leftarrow Z
\]


This is the only DAG relative to which the given probability
distribution satisfies MC and FC.


### 4.5 Identifiability with Assumptions about Functional Form



Suppose we have a SEM with endogenous variables \(\bV\) and exogenous
variables \(\bU\), where each variable in \(\bV\) is determined by an
equation of the form:

 

\[X\_i = f\_i (\bPA(X\_i), U\_i).\]


Suppose, moreover, that we have a probability distribution \(\Pr'\) on
\(\bU\) in which all of the \(U\_i\)s are independent. This will induce
a probability distribution P on \(\bV\) that satisfies MC relative to
the correct causal DAG on \(\bV\). In other words, our probabilistic
SEM will generate a unique causal Bayes net. The methods described in
the previous section attempt to infer the underlying graph \(\bG\)
from relations of probabilistic dependence and independence. These
methods can do no better than identifying the Markov equivalence
class. Can we do better by making use of additional information about
the probability distribution P, beyond relations of dependence and
independence?



There is good news and there is bad news. First the bad news. If the
variables in \(\bV\) are discrete, and we make no assumptions about
the form of the functions \(f\_i\), then we can infer no more about the
SEM than the Markov equivalence to which the graph belongs (Meek
1995).



More bad news: If the variables in \(\bV\) are continuous, the
simplest assumption, and the one that has been studied in most detail,
is that the equations are *linear* with *Gaussian*
(normal, or bell-shaped) errors. That is:


* \(X\_i = \sum\_j c\_j X\_j + U\_i\), where *j* ranges over the
indices of \(\bPA(X\_i)\) and the \(c\_j\)s are constants
* \(Pr'\) assigns a Gaussian distribution to each \(U\_i\)



It turns out that with these assumptions, we can do no better than
inferring the Markov equivalence class of the causal graph on \(\bV\)
from probabilistic dependence and independence (Geiger & Pearl
1988).



Now for the good news. There are fairly general assumptions that allow
us to infer a good deal more. Here are some fairly simple cases:




(**LiNGaM**) (Shimizu et al. 2006)  

If:


1. The variables in \(\bV\) are continuous;
2. The functions \(f\_i\) are linear;
3. The probability distributions on the error variables \(U\_i\) are
not Gaussian (or at most one is Gaussian);
4. The error variables \(U\_i\) are probabilistically independent in
\(\Pr'\);



then the correct DAG on \(\bV\) can be uniquely determined by the
induced probability distribution P on \(\bV\).



(**Non-linear additive**) (Hoyer et al. 2009)  

Almost all functions of the following form allow the correct DAG on
\(\bV\) to be uniquely determined by the induced probability
distribution P on \(\bV\).:


1. The functions \(f\_i\) are nonlinear and the errors are additive
(so \(X\_i = f\_i (\bPA(X\_i)) + U\_i\), with \(f\_i\) nonlinear);
2. The error variables \(U\_i\) are probabilistically independent in
\(\Pr'\);




In fact, this case can be generalized considerably:



(**Post non-linear**) (Zhang & Hyvarinen
2009)  

With the exception of five specific cases that can be fully specified,
all functions of the following form allow the correct DAG on \(\bV\)
to be uniquely determined by the induced probability distribution P on
\(\bV\).:

1. The functions have the form \(X\_i = g\_i (f\_i (\bPA(X\_i)) + U\_i)\)
with \(f\_i\) and \(g\_i\) nonlinear, and \(g\_i\) invertible;
2. The error variables \(U\_i\) are probabilistically independent in
\(\Pr'\);




See also Peters et al. (2017) for discussion.



While there are specific assumptions behind these results, they are
nonetheless remarkable. They entail, for example, that (given the
assumptions of the theorems) knowing only the probability distribution
on two variables *X* and *Y*, we can infer whether *X*
causes *Y* or *Y* causes *X*.


### 4.6 Latent Common Causes



The discussion so far has focused on the case where there are no
latent common causes of the variables in \(\bV\), and the error
variables \(U\_i\) can be expected to be probabilistically independent.
As we noted in
 Section 2.3
 above, we represent a latent common cause with a double-headed arrow.
For example, the acyclic directed mixed graph in
 Figure 9
 represents a latent common cause of *X* and *Z*. More
generally, we can use an ADMG like Figure 9 to represent that the
error variables for *X* and *Z* are not probabilistically
independent.



![diagram: X has an arrow pointing east to Y which in turn has an arrow pointing east to Z; X and Z are connected by a curved double-headed arrow](fig9.svg)

Figure 9




If there are latent common causes, we expect MCScreening\_off and MCFactorization to fail if we apply them in a
naive way. In
 Figure 9,
 *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 on \(\{L, X, Y, Z\}\)
satisfies MC with respect to the DAG that includes *L* as a
common cause of *X* and *Z*. Then it turns out that the
probability distribution will *still* satisfy MC*d*-separation with respect to the ADMG of Figure 9. A causal
model incorporating an ADMG and probability distribution satisfying
MC*d*-separation is called a *semi-Markov
causal model* (SMCM).



If we allow that the correct causal graph may be an ADMG, 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, suppose that the probability
distribution on \(\{X, Y, Z\}\) has the following features:


* *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*



We saw in
 Section 4.4
 that the only DAG that implies just these (in)dependencies is:



\[
X \rightarrow Y \leftarrow Z
\]


But if we allow for the possibility of latent common causes, there
will be additional ADMGs that also imply just these (in)dependencies.
For example, the structure

 

\[
X \leftrightarrow Y \leftrightarrow Z
\]


is also in the Markov equivalence class, as are several others.



Latent variables present a further complication. Unlike the case where
the error variables \(U\_i\) are probabilistically independent, a SEM
with correlated error terms may imply probabilistic constraints in
addition to conditional (in)dependence relations, even in the absence
of further assumptions about functional form. This means that we may
be able to rule out some of the ADMGs in the Markov equivalence class
using different kinds of probabilistic constraints.


### 4.7 Interventions



A conditional probability such as \(\Pr(Y = y \mid X = x)\) 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 (2009) writes \(\Pr(Y =
y \mid \ido(X = x))\) to characterize this probability. The notation
is misleading, since \(\ido(X = x)\) is not an event in the original
probability space. It might be more accurate to write \(\Pr\_{\ido(X =
x)} (Y = y)\), but we will use Pearl's notation here. 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.
Graphically, we can represent the effect of this intervention by
eliminating the arrows directed into the variable intervened upon.
Such an intervention is sometimes described as "breaking"
those arrows. As we saw in Section
 3.1, in the
context of a SEM, we represent an intervention that sets *X* to
*x* by replacing the equation for *X* with a new one
specifying that \(X = x\).



As we saw in
 Section 3.2,
 there is a close connection between interventions and
counterfactuals; in particular, the antecedents of structural
counterfactuals are thought of as being realized by interventions.
Nonetheless, Pearl (2009) distinguishes claims about interventions
represented by the *do* operator from counterfactuals. The former
are understood in the indicative mood; they concern interventions that
are actually performed. Counterfactuals are in the subjunctive mood,
and concern hypothetical interventions. This leads to an important
epistemological difference between ordinary interventions and
counterfactuals: they behave differently in the way that they interact
with observations of the values of variables. In the case of
interventions, we are concerned with evaluating probabilities such
as

 

\[\Pr(\bY = \by \mid \bX =\bx, \ido(\bZ = \bz)).\]


We assume that the intervention \(\ido(\bZ = \bz)\) is being performed
in the actual world, and hence that we are observing the values that
other variables take \((\bX =\bx)\) in the same world where the
intervention takes place. In the case of counterfactuals, we observe
the value of various variables in the actual world, in which there is
no intervention. We then ask what *would* have happened if an
intervention *had been* performed. The variables whose values
we observed may well take on *different* values in the
hypothetical world where the intervention takes place. Here is a
simple illustration of the difference. Suppose that we have a causal
model in which treatment with a drug causes recovery from a disease.
There may be other variables and causal relations among them as
well.




Intervention:



* An intervention was performed to treat a particular patient with
the drug, and it was observed that she did not recover.
* Question: What is the probability that she recovered,
given the intervention and the observed evidence?
* Answer: Zero, trivially.




Counterfactual: 



* It was observed that a patient did not recover from the
disease.
* Question: What is the probability that she would have
recovered, had she been treated with the drug?
* Answer: Nontrivial. The answer is not necessarily
zero, nor is it necessarily P(*recovery* | *treatment*).
If we know that she was in fact treated, then we could infer that she
would not have recovered if treated. But we do not know whether she
was treated. The fact that she did not recover gives us partial
information: it makes it less likely that she was in fact treated; it
also makes it more likely that she has a weak immune system, and so
on. We must make use of all of this information in trying to determine
the probability that she would have recovered if treated.





We will discuss interventions in the present section, and
counterfactuals in
 Section 4.10
 below.



Suppose that we have an acyclic structural equation model with
exogenous variables \(\bU\) and endogenous variables \(\bV\). We have
equations of the form

 

\[X\_i = f\_i (\bPA(X\_i), U\_i),\]


and a probability distribution \(\Pr'\) on the exogenous variables
\(\bU\). \(\Pr'\) then induces a probability distribution P on
\(\bV\). To represent an intervention that sets \(X\_k\) to \(x\_k\), we
replace the equation for \(X\_k\) with \(X\_k = x\_k\). Now \(\Pr'\)
induces a new probability distribution P\* on \(\bV\) (since settings
of the exogenous variables \(\bU\) give rise to different values of
the variables in \(\bV\) after the intervention). P\* is the new
probability distribution that Pearl writes as \(\Pr(* \mid \ido(X\_k
= x\_k))\).



But even if we do not have a complete SEM, we can often compute the
effect of interventions. 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 4.2),
 which tells us:

 

\[\Pr(X\_1, X\_2 , \ldots ,X\_n) = \prod\_i \Pr(X\_i \mid \bPA(X\_i)).\]


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

 

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


where \(\Pr'(X\_k = x\_k) = 1\). The conditional probabilities of the
form \(\Pr(X\_i \mid \bPA(X\_i))\) for \(i \ne k\) remain unchanged by
the intervention. This gives the same result as computing the result
of an intervention using a SEM, when the latter is available. This
result can be generalized to the case where the intervention imposes a
probability distribution \(\Pr^{\dagger}\) on some subset of the
variables in \(\bV\). For simplicity, let's re-label the
variables so that \(\{X\_1, X\_2 ,\ldots ,X\_k\}\) is the set of
variables that we intervene on. Then, the post-intervention
probability distribution is:

 

\[
\Pr^\*(X\_1, X\_2 , \ldots ,X\_n) = \Pr^{\dagger}( X\_1, X\_2 ,\ldots ,X\_k) \times \prod\_{k \lt i \le n} \Pr(X\_i \mid \bPA(X\_i)).
\]


The *Manipulation Theorem* of SGS (2000: theorem 3.6)
generalizes this 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, where the causal
structure on \(\bV\) is represented by an ADMG (including
double-headed arrows) instead of a DAG. The axioms of this system are
presented in
 Supplement on the *do*-calculus.
 One useful special case is given by the




*Back-Door Criterion*. Let *X* and *Y* be variables
in \(\bV\), and \(\bZ \subseteq \bV \setminus \{X, Y\}\) such
that:


1. no member of \(\bZ\) is a descendant of *X*; and
2. every path between *X* and *Y* that terminates with an
arrow into *X* either (a) includes a non-collider in \(\bZ\), or
(b) includes a collider that has no descendants in \(\bZ\);



then \(\Pr(Y \mid \ido(X), \bZ) = \Pr(Y \mid X, \bZ)\).




That is, if we can find an appropriate conditioning set \(\bZ\), the
probability resulting from an intervention on *X* will be the
same as the conditional probability corresponding to an observation of
*X*.


### 4.8. Interventionist Decision Theory



Evidential Decision Theory of the sort developed by Jeffrey (1983),
runs into well-known problems in variants of *Newcomb's
problem* (Nozick 1969). For example, suppose Cheryl believes the
following: She periodically suffers from a potassium deficiency. This
state produces two effects with high probability: It causes her to eat
bananas, which she enjoys; and it causes her to suffer debilitating
migraines. On days when she suffers from the potassium deficiency, she
has no introspective access to this state. In particular, she is not
aware of any banana cravings. Perhaps she rushes to work every
morning, grabbing whatever is at hand to eat on her commute.
Cheryl's causal model is represented by the DAG in
 Figure 10.




![diagram with K having an arrow pointing northwest to B and northeast to M](fig10.svg)




Figure 10




\(K = 1\) represents potassium deficiency, \(B = 1\) eating a banana,
and \(M = 1\) migraine. Her probabilities are as follows:



\[
\begin{aligned}
\Pr(K = 1) & = .2\\
\Pr(B = 1 \mid K = 1) & = .9, &\Pr(B = 1 \mid K = 0) & = .1\\
\Pr(M = 1 \mid K = 1) & = .9, & \Pr(M = 1 \mid K = 0) & = .1
\end{aligned}
\]


Her utility for the state of the world \(w \equiv \{K = k, B = b, M =
m\}\) is \(\Ur(w) = b - 20m\). That is, she gains one unit of utility
for eating a banana, but loses 20 units for suffering a migraine. She
assigns no intrinsic value to the potassium deficiency.



Cheryl is about to leave for work. Should she eat a banana? According
to *Evidential Decision Theory* (EDT), Cheryl should maximize
*Evidential Expected Utility*, where

 

\[\EEU(B = b) = \sum\_w \Pr(w \mid B = b)\Ur(w)\]


From the probabilities given, we can compute that:

 

\[
\begin{aligned}
\Pr(M = 1 \mid B = 1) & \approx .65\\
\Pr(M = 1 \mid B = 0) & \approx .12
\end{aligned}
\]


Eating a banana is strongly correlated with migraine, due to the
common cause. Thus

 

\[\begin{aligned}
\EEU(B = 1) &\approx {-12}\\
\EEU(B = 0) & \approx {-2.4}
\end{aligned}
\]


So EDT, at least in its simplest form, recommends abstaining from
bananas. Although Cheryl enjoys them, they provide strong evidence
that she will suffer from a migraine.



Many think that this is bad advice. Eating a banana does not
*cause* Cheryl to get a migraine; it is a harmless pleasure. A
number of authors have formulated versions of *Causal Decision
Theory* (CDT) that aim to incorporate explicitly causal
considerations (e.g., Gibbard & Harper 1978; Joyce 1999; Lewis
1981; Skyrms 1980). Causal models provide a natural setting for CDT,
an idea proposed by Meek and Glymour (1994) and developed by Hitchcock
(2016), Pearl (2009: Chapter 4) and Stern (2017). The central idea is
that the agent should treat her action as an *intervention.*
This means that Cheryl should maximize her *Causal Expected
Utility*:

 

\[\CEU(B = b) = \sum\_w \Pr(w \mid \ido(B = b))\Ur(w)\]


Now we can compute

 

\[
\begin{aligned}
\Pr(M = 1 \mid \ido(B = 1)) & = .26\\
\Pr(M = 1 \mid \ido(B = 0)) & = .26
\end{aligned}
\]


So that now

 

\[
\begin{aligned}
\CEU(B = 1) &= {-4.2}\\
\CEU(B = 0) & = {-5.2}
\end{aligned}
\]


This yields the plausible result that eating a banana gives Cheryl a
free unit of utility. By intervening, Cheryl breaks the arrow from
*K* to *B* and destroys the correlation between eating a
banana and suffering a migraine.



More generally, one can use the methods for calculating the effects of
interventions described in the previous section to compute the
probabilities needed to calculate Causal Expected Utility. Stern
(2017) expands this approach to allow for agents who distribute their
credence over multiple causal models. Hitchcock (2016) shows how the
distinction between interventions and counterfactuals, discussed in
more detail in
 Section 4.10
 below, can be used to deflect a number of alleged counterexamples to
CDT.



There is much more that can be said about the debate between EDT and
CDT. For instance, if Cheryl knows that she is intervening, then she
will not believe herself to be accurately described by the causal
structure in
 Figure 10.
 Instead, she will believe herself to instantiate a causal structure
in which the arrow from *K* to *B* is removed. In this
causal structure, if P satisfies MC, we will have \(\Pr(w \mid B = b)
= \Pr(w \mid \ido(B = b))\), and the difference between EDT and CDT
collapses. If there is a principled reason why a deliberating agent
will always believe herself to be intervening, then EDT will yield the
same normative recommendations as CDT, and will avoid counterexamples
like the one described above. Price's defense of EDT (Price
1986) might be plausibly reconstructed along these lines. So the moral
is not necessarily that CDT is normatively correct, but rather that
causal models may be fruitfully employed to clarify issues in decision
theory connected with causation.


### 4.9 Causal Discovery with Interventions



In the previous section, we discussed how to use knowledge (or
assumptions) about the structure of a causal graph \(\bG\) to make
inferences about the results of interventions. In this section, we
explore the converse problem. If we can intervene on variables and
observe the post-intervention probability distribution, what can we
infer about the underlying causal structure? This topic has been
explored extensively in the work of Eberhardt and his collaborators.
(See, for example, Eberhardt & Scheines 2007 and Hyttinen et al.
2013a.) Unsurprisingly, we can learn more about causal structure if we
can perform interventions than if we can only make passive
observations. However, just how much we can infer depends upon what
kinds of interventions we can perform, and on what background
assumptions we make.



If there are no latent common causes, so that the true causal
structure on \(\bV\)
is represented by a DAG \(\bG\), then it will always be possible to discover
the complete causal structure using interventions. If we can only
intervene on one variable at a time, we may need to separately
intervene on all but one of the variables before the causal structure
is uniquely identified. If we can intervene on multiple variables at
the same time, we can discover the true causal structure more
quickly.



If there are latent common causes, so that the true causal structure
on \(\bV\) is represented by an ADMG, then it may not be possible to
discover the true causal structure using only single-variable
interventions. (Although we can do this in the special case where the
functions in the underlying structural equation model are all linear.)
However, if we can intervene on multiple variables at the same time,
then it is possible to discover the true causal graph.



Eberhardt and collaborators have also explored causal discovery using
*soft* interventions. A soft intervention influences the value
of a variable without breaking the arrows into that variable. For
instance, suppose we want to know whether increasing the income of
parolees will lead to decreased recidivism. We randomly divide
subjects into treatment and control conditions, and give regular cash
payments to those in the treatment condition. This is not an
intervention on income *per se*, since income will still be
influenced by usual factors: savings and investments, job training,
help from family members, and so on. Soft interventions facilitate
causal inference because they create colliders, and as we have seen,
colliders have a distinct probabilistic signature. Counterintuitively,
this means that if we want to determine whether *X* causes
*Y* it is desirable to perform a soft intervention on *Y*
(rather than *X*), to see if we can create a collider
\(I\rightarrow Y\leftarrow X\) (where *I* is the intervention).
Soft interventions are closely related to *instrumental
variables.* If there are no latent common causes, we can infer the
true causal structure using soft interventions. Indeed, if we can
intervene on every variable at once, we can determine the correct
causal structure from this one intervention. However, if there are
latent common causes, it is not in general possible to discover the
complete causal structure using soft interventions. (Although this can
be done if we assume linearity.)


### 4.10 Counterfactuals



Section 3.3
 above discussed counterfactuals in the context of deterministic
causal models. The introduction of probability adds a number of
complications. In particular, we can now talk meaningfully about the
probability of a counterfactual being true. Counterfactuals play a
central role in the *potential outcome* framework for causal
models pioneered by Neyman (1923), and developed by Rubin (1974) and
Robins (1986), among others.



Counterfactuals in the potential outcome framework interact with
probability differently than counterfactuals in Lewis's (1973b)
framework. Suppose that Ted was exposed to asbestos and developed lung
cancer. We are interested in the counterfactual: "If Ted had not
been exposed to asbestos, he would not have developed lung
cancer". Suppose that the processes by which cancer develops are
genuinely indeterministic. Then, it seems wrong to say that if Ted had
not been exposed to asbestos, he definitely would have developed lung
cancer; and it seems equally wrong to say that he definitely would not
have developed lung cancer. In this case, Lewis would say that the
counterfactual "If Ted had not been exposed to asbestos, he
would not have developed lung cancer" is determinately
*false*. As a result, the objective probability of this
counterfactual being true is zero. On the other hand, a counterfactual
with objective probability *in the consequent* may be true:
"If Ted had not been exposed to asbestos, his objective chance
of developing lung cancer would have been .06". By contrast, in
the potential outcome framework, probability may be pulled out of the
consequent and applied to the counterfactual as a whole: The
probability of the counterfactual "If Ted had not been exposed
to asbestos, he would have developed lung cancer" can be
.06.



If we have a complete structural equation model, we can assign
probabilities to counterfactuals, in light of observations. Let \(\bV
= \{X\_1, X\_2 ,\ldots ,X\_n\}\) be a set of endogenous variables, and
\(\bU = \{U\_1, U\_2 ,\ldots ,U\_n\}\) a set of exogenous variables. Our
structural equations have the form:

 

\[X\_i = f\_i (\bPA(X\_i), U\_i)\]


We have a probability distribution \(\Pr'\) on \(\bU\), which induces
a probability distribution P on \(\bU \cup \bV\). Suppose that we
observe the value of some of the variables: \(X\_j = x\_j\) for all \(j
\in \bS \subseteq \{1,\ldots ,n\}\). We now want to assess the
counterfactual "if \(X\_k\) had been \(x\_k\), then \(X\_l\) would
have been \(x\_l\)", where *k* and *l* may be in
\(\bS\) but need not be. We can evaluate the probability of this
counterfactual using this three-step process:


1. Update the probability P by conditioning on the observations, to
get a new probability distribution \(\Pr(* \mid \cap\_{j \in \bS}
X\_j = x\_j)\). Call the restriction of this probability function to
\(\bU\) \(\Pr''\).
2. Replace the equation for \(X\_k\) with \(X\_k = x\_k\).
3. Use the distribution \(\Pr''\) on \(\bU\) together with the
modified set of equations to induce a new probability distribution P\*
on \(\bV\). \(\Pr^\*( X\_l = x\_l)\) is then the probability of the
counterfactual.



This procedure differs from the procedure for interventions (discussed
in
 Section 4.7)
 in that steps 1 and 2 have been reversed. We first update the
probability distribution, then perform the intervention. This reflects
the fact that the observations tell us about the actual world, in
which the intervention did not (necessarily) occur.



If we do not have a complete SEM, it is not generally possible to
identify the probability of a counterfactual, but only to set upper
and lower bounds. For example, suppose that we believe that asbestos
exposure causes lung cancer, so that we posit a simple DAG:



\[A \rightarrow L.\]


Suppose also that we have data for people similar to Ted which yields
the following probabilities:

 

\[\begin{aligned}
\Pr(L = 1 \mid A = 1) & = .11,\\
\Pr(L = 1 \mid A = 0) & = .06.
\end{aligned}\]


(We are oversimplifying, and treating asbestos and lung cancer as
binary variables.) We observe that Ted was in fact exposed to asbestos
and did in fact develop lung cancer. What is the probability of the
counterfactual: "If Ted had not been exposed to asbestos, he
would not have developed lung cancer"? Pearl (2009) calls a
probability of this form a *probability of necessity.* It is
often called the *probability of causation*, although this
terminology is misleading for reasons discussed by Greenland and
Robins (1988). This quantity is often of interest in tort law. Suppose
that Ted sues his employer for damages related to his lung cancer. He
would have to persuade a jury that his exposure to asbestos caused his
lung cancer. American civil law requires a "more probable than
not" standard of proof, and it employs a "but for"
or counterfactual definition of causation. Hence Ted must convince the
jury that it is more probable than not that he would not have
developed lung cancer if he had not been exposed.



We may divide the members of the population into four categories,
depending upon which counterfactuals are true of them:


* *doomed* individuals will develop lung cancer no matter
what
* *immune* individuals will avoid lung cancer no matter
what
* *sensitive* individuals will develop lung cancer just in
case they are exposed to asbestos
* *reverse sensitive* individuals will develop lung cancer
just in case they are not exposed to asbestos



It is easiest to think of the population as being divided into four
categories, with each person being one of these four types. However,
we do not need to assume that the process is deterministic; it may be
the case that each person only has a certain probability of falling
into one of these categories.



Mathematically, this is equivalent to the following. Let \(U\_L\) be
the error variable for \(L. U\_L\) takes values of the form \((u\_1,
u\_2)\) with each \(u\_i\) being 0 or 1. \((1, 1)\) corresponds to
*doomed*, \((0, 0)\) to *immune*, \((1, 0)\) to
*sensitive*, and \((0, 1)\) to *reverse*. That is, the
first element tells us what value *L* will take if an individual
is exposed to asbestos, and the second element what value *L*
will take if an individual is not exposed. The equation for *L*
will be \(L = (A \times u\_1) + ((1 - A) \times u\_2)\).



Let us assume that the distribution of the error variable \(U\_L\) is
independent of asbestos exposure *A*. The observed probability of
lung cancer is compatible with both of the following probability
distributions over our four counterfactual categories:



\[
\begin{aligned}
\Pr\_1(\textit{doomed}) & = .06, &\Pr\_2(\textit{doomed}) &= 0,\\
\Pr\_1(\textit{immune}) & = .89, & \Pr\_2(\textit{immune}) & = .83,\\
\Pr\_1(\textit{sensitive}) & = .05, & \Pr\_2(\textit{sensitive}) & = .11, \\
\Pr\_1(\textit{reverse}) & = 0 & \Pr\_2(\textit{reverse}) & = .06
\end{aligned}
\]

 

More generally, the observed probability is compatible with any
probability \(\Pr'\) satisfying:

 

\[
\begin{aligned}
\Pr'(\textit{doomed}) + \Pr'(\textit{senstive}) & = \Pr(L \mid A) & = .11;\\
\Pr'(\textit{immune}) + \Pr'(\textit{reverse}) & = \Pr({\sim}L \mid A) & = .89;\\
\Pr'(\textit{doomed}) + \Pr'(\textit{reverse}) & = \Pr(L \mid {\sim}A) & = .06;\\
\Pr'(\textit{immune}) + \Pr'(\textit{senstive}) & = \Pr({\sim}L \mid {\sim}A) & = .94.\\
 \end{aligned}
 \]


\(\Pr\_1\) and \(\Pr\_2\) are just the most extreme cases. From the fact
that Ted was exposed to asbestos and developed lung cancer, we know
that he is either *sensitive* or *doomed*. The
counterfactual of interest will be true just in case he is
*sensitive*. Hence the probability of the counterfactual, given
the available evidence, is P(*sensitive* | *sensitive*
or *doomed*). However, using \(\Pr\_1\) yields a conditional
probability of .45 (5/11), while \(\Pr\_2\) yields a conditional
probability of 1. Given the information available to us, all we can
conclude is that the probability of necessity is between .45 and 1. To
determine the probability more precisely, we would need to know the
probability distribution of the error variable.



A closely related counterfactual quantity is what Pearl (2009) calls
the *probability of sufficiency.* Suppose that Teresa, unlike
Ted, was not exposed to asbestos, and did not develop lung cancer. The
probability of sufficiency is the probability that she *would*
have suffered lung cancer if she *had been* exposed. That is,
the probability of sufficiency is the probability that if the cause
were added to a situation in which it and the effect was absent, it
would have resulted in the effect occurring. The probability of
sufficiency is closely related to the quantity that Sheps (1958)
called the *relative difference*, and that Cheng (1997) calls
the *causal power.* Cheng's terminology reflects the idea
that the probability of sufficiency of *C* for *E* is the
power of *C* to bring about *E* in cases where *E* is
absent. As in the case of the probability of necessity, if one does
not have a complete structural equation model, but only a Causal Bayes
Net or Semi-Markov Causal Model, it is usually only possible to put
upper and lower bounds on the probability of sufficiency. Using the
probabilities from the previous example, the probability of
sufficiency of asbestos for lung cancer would be between .05 (5/94)
and .12 (11/94).



Determining the probabilities of counterfactuals, even just upper and
lower bounds, is computationally demanding. Balke and Pearl's
twin network method (Balke and Pearl (1994a), (1994b); Pearl (2009,
pp. 213 - 215)) and Richardson and Robins' split-node method
(Richardson and Robins (2016)) are two methods that have been proposed
for solving this kind of problem.


## 5. Further Reading



The most important works surveyed in this entry are Pearl 2009 and
Spirtes, Glymour, & Scheines 2000. Pearl 2010, Pearl et al. 2016,
and Pearl & Mackenzie 2018 are three overviews of Pearl's
program. Pearl 2010 is the shortest, but the most technical. Pearl
& Mackenzie 2018 is the least technical. Scheines 1997 and the
"Introduction" of Glymour & Cooper 1999 are accessible
introductions to the SGS program. Eberhardt 2009, Hausman 1999,
Glymour 2009, and Hitchcock 2009 are short overviews that cover some
of the topics raised in this entry.



The entry on
 causation and manipulability
 contains extensive discussion of interventions, and some discussion
of causal models.



Halpern (2016) engages with many of the topics in
 Section 3.
 See also the entry for
 counterfactual theories of causation.



The entry on
 probabilistic causation
 contains some overlap with the present entry. Some of the material from
 Section 4
 of this entry is also presented in Section 3 of that entry. That
entry contains in addition some discussion of the connection between
probabilistic causal models and earlier probabilistic theories of
causation.



Eberhardt 2017 is a short survey that provides a clear introduction to
many of the topics covered in
 Sections 4.2
 through 4.6, as well as Section
 4.9. Spirtes and
Zhang 2016 is a longer and more technical overview that covers much of
the same ground. It has particularly good coverage on the issues
raised in
 Section 4.5.



The entries on
 decision theory
 and
 causal decision theory
 present more detailed background information about some of the issues
raised in
 Section 4.8.



This entry has focused on topics that are likely to be of most
interest to philosophers. There are a number of important technical
issues that have been largely ignored. Many of these address problems
that arise when various simplifying assumptions made here (such as
acyclicity, and knowledge of the true probabilities) are
rejected. Some of these issues are briefly surveyed along with
references in
 Supplement on Further Topics in Causal Inference.