Probabilistic graphical models, such as Markov random fields (MRF), exploit dependencies among random variables to model a rich family of joint probability distributions. Inference algorithms, such as belief propagation (BP), can effectively compute the marginal posteriors for decision making. Nonetheless, inferences involve sophisticated probability calculations and are difficult for humans to interpret. Among all existing explanation methods for MRFs, no method is designed for fair attributions of an inference outcome to elements on the MRF where the inference takes place. Shapley values provide rigorous attributions but so far have not been studied on MRFs. We thus define Shapley values for MRFs to capture both probabilistic and topological contributions of the variables on MRFs. We theoretically characterize the new definition regarding independence, equal contribution, additivity, and submodularity. As bruteforce computation of the Shapley values is challenging, we propose GraphShapley, an approximation algorithm that exploits the decomposability of Shapley values, the structure of MRFs, and the iterative nature of BP inference to speed up the computation. In practice, we propose metaexplanations to explain the Shapley values and make them more accessible and trustworthy to human users. On four synthetic and nine realworld MRFs, we demonstrate that GraphShapley generates sensible and practical explanations.
Correlated Equilibria for Approximate Variational Inference in MRFs
Almost all of the work in graphical models for game theory has mirrored previous work in probabilistic graphical models. Our work considers the opposite direction: Taking advantage of advances in equilibrium computation for probabilistic inference. In particular, we present formulations of inference problems in Markov random fields (MRFs) as computation of equilibria in a certain class of gametheoretic graphical models. While some previous work explores this direction, we still lack a more precise connection between variational probabilistic inference in MRFs and correlated equilibria. This paper sharpens the connection, which helps us exploit relatively more recent theoretical and empirical results from the literature on algorithmic and computational game theory on the tractable, polynomialtime computation of exact or approximate correlated equilibria in graphical games with arbitrary, loopy graph structure. Our work discusses how to design new algorithms with equally tractable guarantees for the computation of approximate variational inference in MRFs. In addition, inspired by a previously stated gametheoretic view of treereweighted messagepassing techniques for belief inference as a zerosum game, we propose a different, generalsum potential game to design approximate fictitiousplay techniques. Empirical evaluations on synthetic experiments and on an application to soft denoising on realworld image datasets illustrate the performance of more »
 Editors:
 Jaeger, Manfred; Nielsen, Thomas Dyhre
 Publication Date:
 NSFPAR ID:
 10294335
 Journal Name:
 Proceedings of Machine Learning Research
 Volume:
 138
 Page Range or eLocationID:
 329  340
 ISSN:
 26403498
 Sponsoring Org:
 National Science Foundation
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