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 — cs-401r:influence-in-bayes-nets [2014/10/13 16:56] (current)cs401rpml created 2014/10/13 16:56 cs401rpml created 2014/10/13 16:56 cs401rpml created Line 1: Line 1: + ==Tutorial: Influence in Bayes Nets== + + ===Arrows=== + 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 $B$". + + ===Conditional Independence=== + 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. + + ===Influence=== + 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 $a$. + + ===Explaining Away=== + 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 $C$.” 