The arrows in a directed graphical model indicate how to factor the joint distribution over the variables in the model. They indicate explicit dependence. You can think of the relationship indicated by an arrow $A \rightarrow B$ as “$A$ causes $B$”.

The “grand unifying principle” for reading influence in a Bayes net (directed graphical model) applies between any pair of nodes in a model: (1) Two nodes (random variables) $A$ & $B$ are conditionally independent, given their parents, if and only if there is no direct arrow between the two nodes (in either direction).

• Note that this statement does not make any guarantees about conditional independence between $A$ & $B$ given other non-parent nodes.

One node (random variable) $A$ is said to influence another node (random variable) $B$ if and only if a conditional query $P(B = b|A = a)$ produces an expression that after algebraic simplification still depends on the value $a$.

In particular, as an example of influence we consider the explaining away structure ($A \rightarrow C$ and $B \rightarrow C$). Here we ask “Does $A$ influence $B$, given $C$?” Since $C$ is not a parent of either $A$ or $B$, then statement (1) does not apply. So, using statement (2), we ask the conditional query $P(B = b|A = a,C = c)$. We do the computation required for a conditional query: first apply the definition of conditional probability, then use marginalization (as necessary) for numerator and for the denominator, then simplify algebraically. Here the meaning of algebraic simplification includes at least factoring out common terms from sums, simplifying sums, and canceling common terms that occur in both numerator and denominator. The result is an expression that does indeed depend on the value $a$, thus the answer to the original question is “Yes, $A$ does influence $B$, given $C$.”