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**1 - 5**of**5**### Revisiting the Limits of MAP Inference by MWSS on Perfect Graphs

"... A recent, promising approach to identifying a configuration of a discrete graphical model with highest probability (termed MAP inference) is to reduce the problem to finding a maximum weight stable set (MWSS) in a derived weighted graph, which, if perfect, allows a solution to be found in polynomial ..."

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A recent, promising approach to identifying a configuration of a discrete graphical model with highest probability (termed MAP inference) is to reduce the problem to finding a maximum weight stable set (MWSS) in a derived weighted graph, which, if perfect, allows a solution to be found in polynomial time. Weller and Jebara (2013) investigated the class of binary pairwise mod-els where this method may be applied. How-ever, their analysis made a seemingly innocuous assumption which simplifies analysis but led to only a subset of possible reparameterizations be-ing considered. Here we introduce novel tech-niques and consider all cases, demonstrating that this greatly expands the set of tractable models. We provide a simple, exact characterization of the new, enlarged set and show how such mod-els may be efficiently identified, thus settling the power of the approach on this class. 1

### 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

### THIS VERSION FIXES A TYPO IN THE STATEMENT OF THEOREM 8 COMPARED TO THE JMLR PUBLISHED VERSION. Revisiting the Limits of MAP Inference by MWSS on Perfect Graphs

"... A recent, promising approach to identifying a configuration of a discrete graphical model with highest probability (termed MAP inference) is to reduce the problem to finding a maximum weight stable set (MWSS) in a derived weighted graph, which, if perfect, allows a solution to be found in polynomial ..."

Abstract
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A recent, promising approach to identifying a configuration of a discrete graphical model with highest probability (termed MAP inference) is to reduce the problem to finding a maximum weight stable set (MWSS) in a derived weighted graph, which, if perfect, allows a solution to be found in polynomial time. Weller and Jebara (2013) investigated the class of binary pairwise mod-els where this method may be applied. How-ever, their analysis made a seemingly innocuous assumption which simplifies analysis but led to only a subset of possible reparameterizations be-ing considered. Here we introduce novel tech-niques and consider all cases, demonstrating that this greatly expands the set of tractable models. We provide a simple, exact characterization of the new, enlarged set and show how such mod-els may be efficiently identified, thus settling the power of the approach on this class. 1

### 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