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Belief propagation on Polytree

I’m working through exercises on Belief Propagation and the Junction Tree Algorithm and I’m stuck with the following problem. Consider the distribution P(A,B,C,D,E,F,G,H)=P(A)P(B)P(C)P(F) P(D|A,B)P(E|B,C)P(G|D,F)P(H|E,F), i.e.

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The questions asked are:

  • Is it possible to use belief propagation on this graph (without forming a Junction Tree) to compute P(B|G,H)?
  • What about P(B|F,G)
  • Now assume only F has been observed. How would you compute P(B|F)?

Now, AFAIK this is a directed polytree (Nodes may have multiple parents, but there is at most a single path between any two nodes). Thus Belief Propagation should allow exact inference in this tree, which is why I would answer "Yes" to both 1) and 2). Is this correct? It seems like I’m missing something here. Why would conditioning change something about whether Belief Propagation can be used here?

Regarding 3, we note that B is cond. indep. of F given no other nodes, so P(B) = P(B|F), is this correct?

Cross Validated Asked by JonaSc on January 2, 2021

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