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1. Correlated Equilibria for Approximate Variational Inference in MRFs

   Ortiz, Luis E.; Wang, Boshen; Gong, Ze (January 2020, Proceedings of Machine Learning Research)

   Jaeger, Manfred; Nielsen, Thomas Dyhre (Ed.)

   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 game-theoretic 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, polynomial-time 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 game-theoretic view of tree-reweighted message-passing techniques for belief inference as a zero-sum game, we propose a different, general-sum potential game to design approximate fictitious-play techniques. Empirical evaluations on synthetic experiments and on an application to soft de-noising on real-world image datasets illustrate the performance of our proposed approach and shed some light on the conditions under which the resulting belief inference algorithms may be most effective relative to standard state-of-the-art methods. 

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2. Shapley Values and Meta-Explanations for Probabilistic Graphical Model Inference

   https://doi.org/10.1145/3340531.3411881  

   Liu, Yifei; Chen, Chao; Liu, Yazheng; Zhang, Xi; Xie, Sihong (October 2020, Proceedings of the 29th ACM International Conference on Information and Knowledge Management (CIKM'2020))

   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 brute-force 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 meta-explanations to explain the Shapley values and make them more accessible and trustworthy to human users. On four synthetic and nine real-world MRFs, we demonstrate that GraphShapley generates sensible and practical explanations. 

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3. Efficient Inference for Untied MLNs

   https://doi.org/10.24963/ijcai.2017/644  

   Sarkhel, Somdeb; Venugopal, Deepak; Ruozzi, Nicholas; Gogate, Vibhav (August 2017, Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence)

   We address the problem of scaling up local-search or sampling-based inference in Markov logic networks (MLNs) that have large shared sub-structures but no (or few) tied weights. Such untied MLNs are ubiquitous in practical applications. However, they have very few symmetries, and as a result lifted inference algorithms--the dominant approach for scaling up inference--perform poorly on them. The key idea in our approach is to reduce the hard, time-consuming sub-task in sampling algorithms, computing the sum of weights of features that satisfy a full assignment, to the problem of computing a set of partition functions of graphical models, each defined over the logical variables in a first-order formula. The importance of this reduction is that when the treewidth of all the graphical models is small, it yields an order of magnitude speedup. When the treewidth is large, we propose an over-symmetric approximation and experimentally demonstrate that it is both fast and accurate. 

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4. The Lauritzen-Chen Likelihood For Graphical Models

   Ilya Shpitser (April 2023, Proceedings of The 26th International Conference on Artificial Intelligence and Statistics)

   Francisco Ruiz, Jennifer Dy (Ed.)

   Graphical models such as Markov random fields (MRFs) that are associated with undirected graphs, and Bayesian networks (BNs) that are associated with directed acyclic graphs, have proven to be a very popular approach for reasoning under uncertainty, prediction problems and causal inference. Parametric MRF likelihoods are well-studied for Gaussian and categorical data. However, in more complicated parametric and semi-parametric set- tings, likelihoods specified via clique potential functions are generally not known to be congenial (jointly well-specified) or non-redundant. Congenial and non-redundant DAG likelihoods are far simpler to specify in both parametric and semi-parametric settings by modeling Markov factors in the DAG factorization. However, DAG likelihoods specified in this way are not guaranteed to coincide in distinct DAGs within the same Markov equivalence class. This complicates likelihoods based model selection procedures for DAGs by “sneaking in” potentially un- warranted assumptions about edge orientations. In this paper we link a density function decomposition due to Chen with the clique factorization of MRFs described by Lauritzen to provide a general likelihood for MRF models. The proposed likelihood is composed of variationally independent, and non-redundant closed form functionals of the observed data distribution, and is sufficiently general to apply to arbitrary parametric and semi-parametric models. We use an extension of our developments to give a general likelihood for DAG models that is guaranteed to coincide for all members of a Markov equivalence class. Our results have direct applications for model selection and semi-parametric inference. 

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5. Negative Weights in Hinge-Loss Markov Random Fields.

   Charles Dickens, Eriq Augustine (July 2021, Workshop on Tractable Probabilistic Modeling (TPM))

   null (Ed.)

   Probabilistic soft logic (PSL) is a framework for instantiating probabilistic graphical models (PGM) representing complex relational data. Weighted first-order logical statements are used as templates for creating potential functions which define the PGM density. Traditionally, PSL constrains weights to be non-negative to ensure maximum a posteriori (MAP) inference is a tractable convex optimization problem. We propose three novel approaches to extending PSL's expressivity to allow negative weights. Notably, we propose the use of Gödel logic for defining potentials from negatively weighted rules. This method improves upon prior work on this topic by preserving both the convexity and scale of the MAP inference problem. Moreover, we show where each of the five methods discussed in this paper overlap and where they most differ. All negative methods are implemented in PSL, and we introduce a tunable synthetic dataset designed to empirically compare the performance of predictions. 

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