An Information Theoretic Measure of Judea Pearl's Identifiability and Causal Influence

In this paper, we define a new information theoretic measure that we call the “uprooted information”. We show that a necessary and sufficient condition for a probability P (s|do(t)) to be “identifiable” (in the sense of Pearl) in a graph G is that its uprooted information be non-negative for all models of the graph G. In this paper, we also give a new algorithm for deciding, for a Bayesian net that is semi-Markovian, whether a probability P (s|do(t)) is identifiable, and, if it is identifiable, for expressing it without allusions to confounding variables. Our algorithm is closely based on a previous algorithm by Tian and Pearl, but seems to correct a small flaw in theirs. In this paper, we also find a necessary and sufficient graphical condition for a probability P (s|do(t)) to be identifiable when t is a singleton set. So far, in the prior literature, it appears that only a sufficient graphical condition has been given for this. By “graphical” we mean that it is directly based on Judea Pearl’s 3 rules of do-calculus.