June 22, 2009

(Jun 12) Meeting Summary: Wainwright / Jordan, ch 4 – 4.1.5 (incl)

Filed under: Uncategorized — umassliving @ 6:01 am
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Discussed the problems with solving the variational form of A(\theta), namely that the space of mean parameters is hard to characterize explicitly and for most distributions, A^*(\mu) does not have an explicit form. Noted that the Bethe Variational Principle (4.16) is an approximation to A(\theta) by utilizing a simple-to-characterize polyhedral outer-bound \mathbb{L}(G) (to approximate \mathbb{M}(G) and applying the Bethe entropy approximation (to approximate A^*(\mu)). Theorem 4.2 shows that a Lagragian method for solving the BVP is equivalent to the sum-product updates, which provides a principled basis for applying the sum-product algorithm to graph with cycles, namely as a means to approximate the Bethe objective function. Note that, as expected, A_{\mathrm{Bethe}}(\theta) = A(\theta) for tree-structured graphs, since sum-product is an exact inference algorithm for tree-structured graphs.

Note that the right hand side of Equation 4.17 should be reversed, \sum \tau_s - 1, to be consistent with a positive \lambda_{SS} in Equation 4.21a.


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