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• W2765431011 abstract "Journal of Machine Learning Research 14 (2013) 3165-3200 Submitted 2/13; Revised 7/13; Published 10/13 Variational Algorithms for Marginal MAP Qiang Liu Alexander Ihler QLIU 1@ UCI . EDU IHLER @ ICS . UCI . EDU Donald Bren School of Information and Computer Sciences University of California, Irvine Irvine, CA, 92697-3425, USA Editor: Amir Globerson Abstract The marginal maximum a posteriori probability (MAP) estimation problem, which calculates the mode of the marginal posterior distribution of a subset of variables with the remaining variables marginalized, is an important inference problem in many models, such as those with hidden vari- ables or uncertain parameters. Unfortunately, marginal MAP can be NP-hard even on trees, and has attracted less attention in the literature compared to the joint MAP (maximization) and marginal- ization problems. We derive a general dual representation for marginal MAP that naturally inte- grates the marginalization and maximization operations into a joint variational optimization prob- lem, making it possible to easily extend most or all variational-based algorithms to marginal MAP. In particular, we derive a set of “mixed-product” message passing algorithms for marginal MAP, whose form is a hybrid of max-product, sum-product and a novel “argmax-product” message up- dates. We also derive a class of convergent algorithms based on proximal point methods, includ- ing one that transforms the marginal MAP problem into a sequence of standard marginalization problems. Theoretically, we provide guarantees under which our algorithms give globally or lo- cally optimal solutions, and provide novel upper bounds on the optimal objectives. Empirically, we demonstrate that our algorithms significantly outperform the existing approaches, including a state-of-the-art algorithm based on local search methods. Keywords: graphical models, message passing, belief propagation, variational methods, maxi- mum a posteriori, marginal-MAP, hidden variable models 1. Introduction Graphical models such as Bayesian networks and Markov random fields provide a powerful frame- work for reasoning about conditional dependency structures over many variables, and have found wide application in many areas including error correcting codes, computer vision, and computa- tional biology (Wainwright and Jordan, 2008; Koller and Friedman, 2009). Given a graphical model, which may be estimated from empirical data or constructed by domain expertise, the term inference refers generically to answering probabilistic queries about the model, such as computing marginal probabilities or maximum a posteriori estimates. Although these inference tasks are NP-hard in the worst case, recent algorithmic advances, including the development of variational methods and the family of algorithms collectively called belief propagation, provide approximate or exact solutions for these problems in many practical circumstances. In this work we will focus on three common types of inference tasks. The first involves maxi- mization or max-inference tasks, sometimes called maximum a posteriori (MAP) or most probable c 2013 Qiang Liu and Alexander Ihler." @default.
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• W2765431011 date "2013-10-01" @default.
• W2765431011 modified "2023-09-24" @default.
• W2765431011 title "Variational Algorithms for Marginal MAP - eScholarship" @default.
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