Results

**1 - 5**of**5**### Uprooting and Rerooting Graphical Models

"... Abstract We show how any binary pairwise model may be 'uprooted' to a fully symmetric model, wherein original singleton potentials are transformed to potentials on edges to an added variable, and then 'rerooted' to a new model on the original number of variables. The new model i ..."

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Abstract We show how any binary pairwise model may be 'uprooted' to a fully symmetric model, wherein original singleton potentials are transformed to potentials on edges to an added variable, and then 'rerooted' to a new model on the original number of variables. The new model is essentially equivalent to the original model, with the same partition function and allowing recovery of the original marginals or a MAP configuration, yet may have very different computational properties that allow much more efficient inference. This metaapproach deepens our understanding, may be applied to any existing algorithm to yield improved methods in practice, generalizes earlier theoretical results, and reveals a remarkable interpretation of the triplet-consistent polytope.

### Bethe and Related Pairwise Entropy Approximations

"... For undirected graphical models, belief propaga-tion often performs remarkably well for approxi-mate marginal inference, and may be viewed as a heuristic to minimize the Bethe free energy. Fo-cusing on binary pairwise models, we demon-strate that several recent results on the Bethe ap-proximation ma ..."

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For undirected graphical models, belief propaga-tion often performs remarkably well for approxi-mate marginal inference, and may be viewed as a heuristic to minimize the Bethe free energy. Fo-cusing on binary pairwise models, we demon-strate that several recent results on the Bethe ap-proximation may be generalized to a broad fam-ily of related pairwise free energy approxima-tions with arbitrary counting numbers. We ex-plore approximation error and shed light on the empirical success of the Bethe approximation. 1

### SUMMARY

"... Belief propagation may be viewed as a heuristic to optimize the Bethe free energy FB, and often performs strikingly well. Here we focus on binary pairwise MRFs, and generalize important results on the Bethe approximation to the broad family of related pairwise entropy approximations with arbitrary c ..."

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Belief propagation may be viewed as a heuristic to optimize the Bethe free energy FB, and often performs strikingly well. Here we focus on binary pairwise MRFs, and generalize important results on the Bethe approximation to the broad family of related pairwise entropy approximations with arbitrary counting numbers. We make observations that shed light on the success of the Bethe approximation. KEY RESULTS, FOR ANY COUNTING NUMBERS All extend earlier results that were specifically for the Bethe approximation: •Given singleton marginals, we provide an analytic solution for optimum pairwise pseudomarginals. •Thus, the approximate free energy FA may be considered a function only of singleton pseudomarginals {qi = q(Xi = 1)}. •We provide upper and lower bounds on first derivatives ∂FA∂qi as a function of qi, that hold for all values of other marginals {qj: j 6 = i}, see Figure. •We use these derivative bounds to construct an -sufficient mesh over pseudomarginals such that the optimum of FA on the mesh is guaranteed to have value within of the global optimum, see Figures. •We derive all second derivatives of the approximate free energy ∂2FA∂qi∂qj. •Using second derivatives, we show that for attractive models, the discrete optimization problem is submodular, hence may be solved efficiently leading to a FPTAS for the approximate log-partition function log ZA (extends to balanced models). BACKGROUND We consider a binary pairwise model with variables V and edges E. p(x) = e−E (x)

### Motivation: undirected graphical models Example: Part of epinions social network (mixed)

, 2015

"... Motivation: undirected graphical models Powerful way to represent relationships across variables Many applications including: computer vision, social network analysis, deep belief networks, protein folding... In this talk, focus on binary pairwise (Ising) models ..."

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Motivation: undirected graphical models Powerful way to represent relationships across variables Many applications including: computer vision, social network analysis, deep belief networks, protein folding... In this talk, focus on binary pairwise (Ising) models

### Methods for Inference in Graphical Models

, 2014

"... Graphical models provide a flexible, powerful and compact way to model relationships between random variables, and have been applied with great success in many domains. Combining prior beliefs with observed evidence to form a prediction is called inference. Problems of great interest include finding ..."

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Graphical models provide a flexible, powerful and compact way to model relationships between random variables, and have been applied with great success in many domains. Combining prior beliefs with observed evidence to form a prediction is called inference. Problems of great interest include finding a configuration with highest probability (MAP inference) or solving for the distribution over a subset of variables (marginal inference). Further, these methods are often critical subroutines for learning the relationships. However, inference is computationally intractable in general. Hence, much effort has focused on two themes: finding subdomains where exact inference is solvable efficiently, or identifying approximate methods that work well. We ex-plore both these themes, restricting attention to undirected graphical models with discrete variables. First we address exact MAP inference by advancing the recent method of reducing the problem to finding a maximum weight stable set (MWSS) on a derived graph, which, if perfect, admits poly-nomial time inference. We derive new results for this approach, including a general decomposition theorem for models of any order and number of labels, extensions of results for binary pairwise models with submodular cost functions to higher order, and a characterization of which binary pair-wise models can be efficiently solved with this method. This clarifies the power of the approach on