Are nodes outside the markov blanket unconditionally independent?

Question

Apologies if my question is deeply flawed, I've been working through a lot of material in the past few weeks and have a few blind spots here and there.

On one level my question is this - given a bayesian network with all directed edges, what is the set of nodes X and Y such that P(Y) = P(Y|X)

My intuition says that this is all Y,X such that Y not in the markov blanket of X. My rationale is that when performing marginalization, for any given term in the marginalization sum, values have been chosen for every node in the markov blanket, rendering the terms chosen for nodes outside the markov blanket irrelevant.

Am I mistaken?

Much thanks.

Answer 1

Consider the bayesian network: X -> Y -> Z

$X$ is out of the Markov blanket of $Z$ (which is $\{Y,Z\}$), yet $Z$ (unconditionally) depends on $X$.

On the other hand, d-separation is what you are looking for. From Pearl's Causality book, you can read:

d-separation is a criterion for deciding, from a given a causal graph, whether a set X of variables is independent of another set Y, given a third set Z.

You can use an empty conditionning set to check whether two variables are (unconditionally) independent. The Bayes-Ball algorithm is a practical algorithm based on this criterion for deciding whether or not two variables are independent.