June 22, 2009

(Jun 19) Meeting Summary: Wainwright / Jordan, ch 4.1.6 – 4.2 (incl)

Filed under: Uncategorized — umassliving @ 6:06 am
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Things to be thinking about:

  1. Ideas for paper(s) to read week of June 29 after finishing Chapter 4. Can be theoretical paper going more in depth on topics covered in Chapter 4 (e.g. read paper on generalized BP), theoretical paper extending Chapter 4 (e.g. Sudderth paper on Bethe approximation as lower bound in certain attractive graphical models), or application paper (e.g. vision paper applying generalized BP).
  2. Ideas for topics to cover in fall semester. Some potential ideas: MCMC methods for inference, learning theory.

Meeting notes:

The reparameterization 4.27 in Proposition 4.3 is only possible because of the overcomplete set of indicator function sufficient statistics. Does using an even more redundant set of sufficient statistics lead to the existence of more reparameterizations and hence more local optima? The Bethe approximation does not give a lower or upper bound in general, unlike the mean field approximation, but in certain cases it can be convexified (Wainwright [246]) to give an upper bound, and in some models it is a lower bound (Sudderth [224]).

Clarified definition of hypergraphs, and noted that edges in hypergraph diagrams (e.g. Fig. 4.4) denote subset inclusion and are used for generalized belief propagation, they are not hyperedges themselves.

Using 4.41, we can see 4.42 can be put into a similar form as the distribution given by the junction tree, 2.12 on p32, by multiplying and dividing by \phi_h as necessary to get the marginals \mu_h for the maximal hyperedges in the numerator and \mu_h for the intersections between maximal hyperedges (separator sets in the junction tree) in the denominator.

Be able to derive 4.45 from 4.44 and 4.42.

Generalized BP is to the Kikuchi approximation as ordinary sum product is to the Bethe approximation. Kikuchi gives a better approximation to both the set of realizable mean parameters M and to the conjugate dual (negative entropy) A*.

On p103, in the center equation array, middle line, \omega(e,f) should be \omega(f,e).


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