# UMass LIVING

## July 9, 2009

### (Jul 9 3-5 PM) Meeting Summary – Chapter 5: Mean Field Methods

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Mean field methods differ from those discussed in chapter 4 in that $\mathcal{M}$ is approximated used a tractable (nonconvex) inner bound for which $A^*$ is easy to characterize. It uses the notion of a tractable subgraph to restrict the set of sufficient statistics by setting their respective canonical parameters to $0$. Setting certain canonical parameters to zero is equivalent to adding certain constraints between the mean parameters (which have an obvious form for the naive mean field case).

We discussed the notions of tractable subgraphs, the lower bound property of mean field, and its interpretations as maximizing the bound and minimizing the KL divergence (the reverse form of that used in EP). Worked through example 5.2 for the naive mean field and example 5.4 for showing the nonconvexity of the objective function. We worked out $\mathbb{P}[X_1 = X_2]$ to show why the fraction in $\theta(q)$ must be 1/2 instead of 1/4. In example 5.4, clarified the difference between convex sets and convex functions and how figure 5.2(b) is depicting sets. Discussed the extreme points of $\mathcal{M}$, how they arise, and why they must be contained in $\mathcal{M}_{F}$, which explains why $\mathcal{M}_{F} \subset \mathcal{M}$ cant be convex. Discussed structured mean field as a way to use arbitrary tractable subgraphs as opposed to just disconnected subgraphs (as in the mean field case). The constraints between the mean parameters take a form that depends on the subgraph and thus is represented generically as $g_{\alpha}$.

Next time we will briefly make sure that everyone understood examples 5.5 and 5.6.