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 {\it 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 {\it 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.