The Causal Markov Condition: Should you choose to accept it?

1
The Causal Markov ConditionShould you choose to
accept it?

• Karen R. Zwier
• Department of History and Philosophy of Science
• University of Pittsburgh

2
The Causal Markov Condition
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• 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 wont hold between arbitrary G
  and P.
• Therefore, criticisms that pick out
  counterexamplespairs 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.

3
Interesting results of counterexample
criticisms
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• Cyclical graphs
• Causal insufficiency
• Logical relationships among variables
• Selection bias / Sampling bias
• Inter-Unit Causation
• Non-homogeneous populations

4
Where were going
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• 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

5
Humes problem
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• 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.

6
Humes problem gets worse
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• 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.
• 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.

7
So now what?
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• Humes argument does not prove that causation is
  not real (i.e. is only an artifact of our minds).
  It only proves that we cant be certain that it
  is real.
• So if, even in the face of Humes 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?

8
The Causal Markov Condition
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• 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).

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Breaking down the CMC
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• The vertex set V \ (Descendants(W) ? Parents(W))
  can be partitioned into the following vertex
  sets
• NPA(W) All non-parental ancestors of W that are
  not also in Descendants(W)
• Siblings(W) All siblings of W that are not also
  in Descendants(W) ? Parents(W)
• Co-Ancestors(W) All ancestors A of any vertex D
  in Descendants(W), where A is not in
  Descendants(W) ? Ancestors(W) ? Siblings(W)
• UnrelatedExogenous(W) All exogenous vertices in
  the graph that are not also in Ancestors(W) ?
  Co-Ancestors(W) and
• 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).

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Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion
11
Breaking down the CMC
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion
?
?

• W V \ (Descendants(W) ? Parents(W))
  Parents(W) entails that
• W NPA(W) Parents(W)
• W Siblings(W) Parents(W)
• W Co-Ancestors(W) Parents(W)
• W UnrelatedExogenous(W) Parents(W)
• W OtherDescendants(W) Parents(W)

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So what would a pragmatic decision to use/not use
the CMC look like?
Introduction Critique of Metaphysical
Approach Pragmatic Approach
Specifics Conclusion
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• 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!

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W NPA(W) Parents(W)
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion
?
?

• 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.

14
Keep/Drop Robustness?
Introduction Critique of Metaphysical
Approach Pragmatic Approach
Specifics Conclusion
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• 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

15
Example
Introduction Critique of Metaphysical
Approach Pragmatic Approach
Specifics Conclusion
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• 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 modelto
  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.

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W Siblings(W) Parents(W)
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion
?
?

• This assumption is Reichenbachs 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.

17
Keep/Drop PCC? Example.
Introduction Critique of Metaphysical
Approach Pragmatic Approach
Specifics Conclusion
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• 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.

18
Conclusions
Introduction Critique of Metaphysical
Approach CMC Breakdown Pragmatic
Approach Conclusion

• Metaphysical debate over the CMC gets us nowhere,
  because we dont 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 worsethe
  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