Analysis of the binary instrumental variable model
Bastian Steudel and Nihat Ay (Max Planck Institute for Mathematics in the Sciences, Germany )
We assume that observational knowledge about a system is given in
terms of a joint probability distribution P(S) of a set of discrete random variables S. What can be said about the family B(S) of possible causal explanations
in terms of Bayesian networks that are consistent with the observation? Here,
we call a Bayesian network consistent if it contains S and its joint probability
distribution is equal to P(S) after marginalization.
The problem of characterizing B(S) in terms of conditional independencies
among the variables S has been investigated thoroughly since the beginning
of causality theory. We account additionally for the degree of stochastic dependence using information theory and are able to extract further information
about B(S). In particular we present a result stating that whenever the mutual
information among variables of S exceeds a certain bound dependent on a natural number c, then in any network B in B(S) there exist c nodes of S that have
a common root. This generalizes a 'qualitative' property of Bayesian networks,
namely that two dependent variables must have a common ancestor, which is a
formalization of Reichenbach's Principle of Common Cause.