Prove proposition 11.7. Proposition 11.7 Let Q(X ) = 1/ZQ j ? j (C j ) be a cluster mean field…

Prove proposition 11.7.

Proposition 11.7

Let Q(X ) = 1/ZQ j ψj (Cj) be a cluster mean field approximation to a set of factors PΦ, and let ψj be a factor of Q. Then, the set Dj of theorem can be written as

In words, this result states that the interfaces of a cluster are simply the places where the cluster scope intersects potentials in Φ that are not fully contained in the cluster. In our grid example, when we choose the clusters to be the individual columns, the interfaces are the intersections with the row potentials, which are precisely the singleton variables that we discussed.

 

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