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Causal Markov condition

The Markov condition, also known as the Markov assumption, is fundamental in Bayesian probability theory. It states that every node in a Bayesian network is conditionally independent of its nondescendants, given its parents, implying no influence on non-descendant nodes. In a DAG, this local condition aligns with the global Markov condition, where d-separations represent conditional independencies. Additionally, each node is independent of the network, given its Markov blanket. The related Causal Markov condition asserts independence given a node’s direct causes, linking the network’s structure to causality. While the Markov condition can hold without causality, equivalence arises only if the network accurately depicts causal relations.

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Motivation

Main article: Probabilistic causation

Statisticians are enormously interested in the ways in which certain events and variables are connected. The precise notion of what constitutes a cause and effect is necessary to understand the connections between them. The central idea behind the philosophical study of probabilistic causation is that causes raise the probabilities of their effects, all else being equal.

A deterministic interpretation of causation means that if A causes B, then A must always be followed by B. In this sense, smoking does not cause cancer because some smokers never develop cancer.

On the other hand, a probabilistic interpretation simply means that causes raise the probability of their effects. In this sense, changes in meteorological readings associated with a storm do cause that storm, since they raise its probability. (However, simply looking at a barometer does not change the probability of the storm, for a more detailed analysis, see:4).

Implications

Dependence and Causation

It follows from the definition that if X and Y are in V and are probabilistically dependent, then either X causes Y, Y causes X, or X and Y are both effects of some common cause Z in V.5 This definition was seminally introduced by Hans Reichenbach as the Common Cause Principle (CCP) 6

Screening

It once again follows from the definition that the parents of X screen X from other "indirect causes" of X (parents of Parents(X)) and other effects of Parents(X) which are not also effects of X.7

Examples

In a simple view, releasing one's hand from a hammer causes the hammer to fall. However, doing so in outer space does not produce the same outcome, calling into question if releasing one's fingers from a hammer always causes it to fall.

A causal graph could be created to acknowledge that both the presence of gravity and the release of the hammer contribute to its falling. However, it would be very surprising if the surface underneath the hammer affected its falling. This essentially states the Causal Markov Condition, that given the existence of gravity the release of the hammer, it will fall regardless of what is beneath it.

See also

Notes

References

  1. Geiger, Dan; Pearl, Judea (1990). "On the Logic of Causal Models". Machine Intelligence and Pattern Recognition. 9: 3–14. doi:10.1016/b978-0-444-88650-7.50006-8. /wiki/Doi_(identifier) ↩

  2. Lauritzen, S. L.; Dawid, A. P.; Larsen, B. N.; Leimer, H.-G. (August 1990). "Independence properties of directed markov fields". Networks. 20 (5): 491–505. doi:10.1002/net.3230200503. /wiki/Doi_(identifier) ↩

  3. Hausman, D.M.; Woodward, J. (December 1999). "Independence, Invariance, and the Causal Markov Condition" (PDF). British Journal for the Philosophy of Science. 50 (4): 521–583. doi:10.1093/bjps/50.4.521. http://philosophy.wisc.edu/hausman/papers/bjps.pdf ↩

  4. Pearl, Judea (2009). Causality. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511803161. ISBN 9780511803161. 9780511803161 ↩

  5. Hausman, D.M.; Woodward, J. (December 1999). "Independence, Invariance, and the Causal Markov Condition" (PDF). British Journal for the Philosophy of Science. 50 (4): 521–583. doi:10.1093/bjps/50.4.521. http://philosophy.wisc.edu/hausman/papers/bjps.pdf ↩

  6. Reichenbach, Hans (1956). The Direction of Time. Los Angeles: University of California Press. ISBN 9780486409269. {{cite book}}: ISBN / Date incompatibility (help) 9780486409269 ↩

  7. Hausman, D.M.; Woodward, J. (December 1999). "Independence, Invariance, and the Causal Markov Condition" (PDF). British Journal for the Philosophy of Science. 50 (4): 521–583. doi:10.1093/bjps/50.4.521. http://philosophy.wisc.edu/hausman/papers/bjps.pdf ↩