Belief Propagation: An Extremely Rudimentary Discussion McLean & Pavel

The Problem When we have a tree structure describing dependencies between variable, we want to update our probability distributions based on evidence Trees are nice to work with, as the probability distribution can be expressed as the product of edge marginals divided by the product of separator node marginals Simple case: if risk of heart disease depends on your father’s risk of heart disease, what can you say about your own risk if you know your grandfather’s risk? Essentially, given a prior distribution on the tree, find the posterior distribution given some observed evidence

The Solution Belief Propagation is an algorithm to incorporate evidence into a tree distribution Non-iterative method: it only requires two passes through the tree to get an updated distribution

The Algorithm First, incorporate evidence: for each observed variable, find one edge that it is a part of, and set all entries in the edge table that do not correspond to the observed value to zero Next, choose some edge as a root Collect evidence in from every direction to the root Normalize the root edge table Distribute evidence out in every direction from the root

Okay, but what do you mean by “collect evidence”? Well, we want to propagate evidence through the system This is fairly simple for singly-linked items: just update marginal based on joint, then update the next joint based on that marginal, and so on: So if we observed x, Txy becomes T*xy, then we get T*y by summing T*xy over all x Then, T*yz = (Tyz)(T*y/Ty) z x y

And if we have multiply linked items? Then it’s slightly (but only slightly) more complicated: Now if we observe x1 and x2, we get T*x1y and T*x2y We then calculate T1y and T2y (the equivalents of T*y from before, but each using only the information from one of the Xs) Now, T*yz=(Tyz)(T1y/Ty)(T2y/Ty) See Pavel’s handout for a complete workthrough using this graph, and a justification of the calculation of T*yz x1 y z x2

But you’ve got two different marginals for Y! That can’t be right! Patience. All will work out in time. After we have finished collecting evidence, we normalize our root table – in this case, the root would be T*yz Now we distribute evidence – this is the same process as collecting evidence, but in the opposite direction Note that now we are only ever having a single “input” edge going to any given node, so we can come up with proper marginals When we’ve finished distributing evidence, we will have a probability distribution over the tree that reflects the incorporated evidence.