The Causal Markov Condition:
Should you choose to accept it?
Karen R. Zwier
Department of History and Philosophy of Science
University of Pittsburgh
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
The Causal Markov Condition
• SGS [1993, 2000] Formulation:
Let G be a causal graph with vertex set V and P be a probability
distribution over the vertices in V generated by the causal
structure represented by G. G and P satisfy the Causal Markov
Condition if and only if for every W in V, W is independent of
V \ (Descendants(W) Parents(W)) given Parents(W). The
debate over the Causal Markov Condition (CMC) has largely
taken place at the logical/metaphysical level
• From the definition above, it should be obvious that this relation
won’t hold between arbitrary G and P.
• Therefore, criticisms that pick out “counterexamples”—pairs of G
and P for which the CMC does not hold, are not actually criticisms
of the CMC.
• These are criticisms of naïve use of the CMC. And they make
known to us interesting situations in which statistical modeling
decisions affect the applicability of the CMC.
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
Interesting results of
“counterexample” criticisms:
• Cyclical graphs
• Causal insufficiency
• Logical relationships among variables
• Selection bias / Sampling bias
• Inter-Unit Causation
• Non-homogeneous populations
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
Where we’re going…
• There is another type of criticism against the CMC:
what I call metaphysical criticism. The debate over the
Causal Markov Condition (CMC) has largely taken
place at the logical/metaphysical level.
• My claim: The validity of the CMC cannot be decided
on a metaphysical basis
• My alternative: pragmatic, material considerations
should decide use/non-use of the CMC
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
Hume’s problem
• Causation is a non-logical, non-conceptual dependence.
Therefore, there is nothing in the concepts of the related
objects that tells us that one causes the other.
• Only objects are observable; causation is not.
• Even if we allow that causation, or a “causal power”
was operative in a certain situation, we still cannot
extend this assumption to future instances because of
the general problem of induction.
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
Hume’s problem gets worse
1. Hume did not consider concepts to be problematic.
For Hume, sense data automatically turns into an idea.
– But concepts are problematic, especially in a scientific
discussion of causation, where our everyday notions and
sense data may not map on to the entities of our theories. The
decision of which variables to consider is not trivial. And
there are many other non-trivial modeling decisions as well.
2. For Hume, “necessary connection” is essential to
causation.
– But in our framework, causation is not limited to necessary
connection. We want to accommodate a probabilistic notion
of causation as well. But what is the connection between
probability and causation? This is what is under debate in the
CMC.
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
So now what?
• Hume’s argument does not prove that causation is not real (i.e. is only an
artifact of our minds). It only proves that we can’t be certain that it is
real.
• So if, even in the face of Hume’s argument, we choose to be realists about
causation (and I do!), we still must learn from Hume and take our
epistemic limitations seriously.
• Specifically, because of all of the diverse modeling possibilities I showed
in the last slide, we cannot make we cannot make inference decisions (i.e.
assumptions about the connection between probability and causation) on
a metaphysical basis. We must make these decisions on a pragmatic
basis, using the material considerations of the situation at hand, after we
have already made data collection decisions.
− Data collection decisions: What units? What variables? What
possible values for those variables? What population? How to
sample?
− Inference decisions: How do we go from our data to a causal
hypothesis? Specifically, what connection should we assume
between causation and probability?
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
The Causal Markov Condition
• …is a potential assumption about the connection
between causation and probability: specifically, an
assumption about the relationship between a causal
graph and the probability distribution over its
variables.
• SGS [1993, 2000] Formulation:
Let G be a causal graph with vertex set V and P be a
probability distribution over the vertices in V
generated by the causal structure represented by G.
G and P satisfy the Causal Markov Condition if and
only if for every W in V, W is independent of
V \ (Descendants(W) Parents(W)) given
Parents(W).
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
Breaking down the CMC
The vertex set V \ (Descendants(W) Parents(W)) can be partitioned
into the following vertex sets:
1. NPA(W): All non-parental ancestors of W that are not also in
Descendants(W);
2. Siblings(W): All siblings of W that are not also in
Descendants(W) Parents(W);
3. Co-Ancestors(W): All ancestors A of any vertex D in
Descendants(W), where A is not in Descendants(W)
Ancestors(W) Siblings(W);
4. UnrelatedExogenous(W): All exogenous vertices in the graph
that are not also in Ancestors(W) Co-Ancestors(W); and
5. OtherDescendants(W): All descendants D of any vertex in
NPA(W) Siblings(W) Co-Ancestors(W)
UnrelatedExogenous(W), where D is not in
Descendants(W) Ancestors(W) Siblings(W) Co-
Ancestors(W).
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
V1 V2 V3 V4 V5
V6 V7 V8 V9 V10
V11 V12 V13 V14 V15
V16 V17 V18 W V19 V20
V21 V22 V23 V24 V25
= Parents(W)
= Descendants(W)
V26
= NPA(W)
= Siblings(W)
V27 V28 V29 V30
= Co-Ancestors(W)
= UnrelatedExogenous(W)
V31
= OtherDescendants(W)
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
Breaking down the CMC
W V \ (Descendants(W) Parents(W)) | Parents(W) entails that:
1.
W NPA(W) | Parents(W)
2.
W Siblings(W) | Parents(W)
3.
W Co-Ancestors(W) | Parents(W)
4.
W UnrelatedExogenous(W) | Parents(W)
5.
W OtherDescendants(W) | Parents(W)
Introduction • Critique of Metaphysical Approach • • Pragmatic Approach • • • Specifics • • • • Conclusion
Introduction • • •• •Critique of Metaphysical Approach • • ••CMC Breakdown • • • Pragmatic Approach • •Conclusion
So what would a pragmatic decision
to use/not use the CMC look like?
• On the basis of the modeling decisions we have made
in the data-gathering phase (e.g. units, variables, etc.)
we may or may not want to assume all of the
conditional independence statements made by the
CMC.
• We can decide to assume a subset of these!
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
W NPA(W) | Parents(W)
This assumption is called robustness.
• The concept of robustness comes from a common way of
understanding physical causation, in which the set of
circumstances immediately preceding an effect is enough to
determine that effect.
• Robustness between a variable A and another variable B means
that B is unaffected by small disturbances in how A comes about
• Given the parents (i.e. direct causes) of a variable W, the non-
parental ancestors (NPA(W)) have no influence whatsoever on the
value of W. Only the direct causes of a vertex W in the graph have
a “special” causal power over W.
Introduction • Critique of Metaphysical Approach • • Pragmatic Approach • • • Specifics • • • • Conclusion
Introduction • • •• •Critique of Metaphysical Approach • • ••CMC Breakdown • • • Pragmatic Approach • •Conclusion
Keep/Drop Robustness?
• Robustness is a standard that, although desirable for reductive
physical accounts, can be difficult to satisfy: it says that for every
variable W in V, we have a complete set of direct causes that
screens off all other ancestors.
• But sometimes this assumption is not necessary for our
purposes…
Introduction • Critique of Metaphysical Approach • • Pragmatic Approach • • • Specifics • • • • Conclusion
Introduction • • •• •Critique of Metaphysical Approach • • ••CMC Breakdown • • • Pragmatic Approach • •Conclusion
Example
• I am buying a tennis racquet. In order to inform my choice, I
would like to know something about the causal relationship
between the price of a tennis racquet (P) is a cause of tennis-
playing success (S).
Setting 1: My goal is simply to find out if P is a cause of S, so I can better
my tennis playing. I may consider other variables as well, but I have
no desire to fine-tune my causal model—to find out if P is a
necessary member of a set of direct causes of S, or a necessary
member of a set of direct causes of one of the ancestors of S.
Here, do not assume robustness in inferring causal graph.
Setting 2: My goal is to maximize the success of my tennis playing while
expending as little effort as possible to “intervene” on my condition.
I want to know about the precise relationship between P and S
within a network of other variables, so that if another set of variables
screens off P, I will no longer worry about the price of my tennis
racquet.
Here, assume robustness in inferring causal graph.
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
W Siblings(W) | Parents(W)
This assumption is Reichenbach’s Principle of the Common
Cause:
“…the common cause is the connecting link which
transforms an independence into a dependence.”
• One goal we often have in science is to separate
phenomena into independent realms so that we can
study them more accurately.
Introduction • Critique of Metaphysical Approach • • Pragmatic Approach • • • Specifics • • • • Conclusion
Introduction • • •• •Critique of Metaphysical Approach • • ••CMC Breakdown • • • Pragmatic Approach • •Conclusion
Keep/Drop PCC? Example.
• The Principle of the Common Cause is controversial particularly
in the context of EPR correlations.
Setting 1: We mean to emphasize that entangled particles are not
independent of each other (and in fact, they are perfectly anti-
correlated). Here, we might choose to represent the measured spin
of each of the particles with a separate variable and discard the
principle of the common cause, allowing a correlation to exist
between the effects.
Do not assume PCC when inferring the causal graph.
Setting 2: We mean to emphasize the separable variables of the system.
Since the two entangled particles are never separable in their
recorded measurements (as far as we know), we might choose to
represent the two measurements together in one variable.
Here, we assume the PCC when inferring the causal graph.
Introduction • • • Critique of Metaphysical Approach • • • CMC Breakdown • • • Pragmatic Approach • • • Conclusion
Conclusions
• Metaphysical debate over the CMC gets us nowhere,
because we don’t have the necessary epistemic access
to the nature of causation
• We can break the CMC down into its component
conditional independence statements and pick and
choose from them in a given situation
• Note: A weaker assumption means that the
underdetermination problem is worse—the hypothesis
space is increased. But there is a trade-off: an
assumption that is too strong for our purpose may
eliminate the very hypothesis that we want to
consider.
• A job for the future: formulating the algorithms based
on weakened CMC assumptions