Approximating the Sum Operation for Marginal-MAP Inference

We study the marginal-MAP problem on graphical models, and present a novel approximation method based on direct approximation of the sum operation. A primary difficulty of marginal-MAP problems lies in the non-commutativity of the sum and max operations, so that even in highly structured models, marginalization may produce a densely connected graph over the variables to be maximized, resulting in an intractable potential function with exponential size. We propose a chain decomposition approach for summing over the marginalized variables, in which we produce a structured approximation to the MAP component of the problem consisting of only pairwise potentials. We show that this approach is equivalent to the maximization of a specific variational free energy, and it provides an upper bound of the optimal probability. Finally, experimental results demonstrate that our method performs favorably compared to previous methods. Introduction Graphical models provide an explicit and compact representation for probability distributions that exhibit factorization structure. They are powerful tools for modeling uncertainty in the field of artificial intelligence, computer vision, bioinformatics, signal processing, and many others. Many such applications can be reduced to basic probabilistic inference tasks; typical tasks include computing marginal probabilities (sum-inference), finding the maximum a posteriori (MAP) estimate (max-inference) and marginal-MAP inference (max-sum-inference). The marginal-MAP problem first marginalizes over a subset of the variables (sum operation), and then seeks the MAP estimate for the rest of the model variables (max operation). Marginal-MAP inference is NP -complete, and harder than either max-inference or sum-inference (Park and Darwiche 2004). Part of the difficulty of marginal-MAP inference lies in the non-commutativity of the sum and the max operations, which can prevent “efficient” elimination orders; even for tree-structured graphical models, it can be computationally intractable (Park and Darwiche 2004; Koller and Friedman 2010). Copyright c © 2012, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. There has been relatively little work on approximating the marginal-MAP problem until recently. State-ofthe-art methods include sampling methods, search methods and message passing methods. Doucet, Godsill, and Robert (2002) propose a simple Markov chain Monte Carlo (MCMC) strategy for marginal-MAP estimates. Johansen, Doucet, and Davy (2008) sample from a sequence of artificial distributions using a sequential Monte Carlo approach. de Campos, Gámez, and Moral (1999) present a genetic algorithm to perform marginal-MAP inference. Park and Darwiche (2004) investigate belief propagation for the approximate sum-inference, and use local search for the approximate max-inference. Huang, Chavira, and Darwiche (2006) propose a branch-and-bound search method for exact marginal-MAP inference by computing the bounds on a compiled arithmetic circuit representation. Dechter and Rish (2003) propose a mini-bucket scheme for the marginal-MAP problem by partitioning the potentials into groups during elimination, and Meek and Wexler (2011) propose a related approximate variable elimination scheme that directly approximates the results of each elimination with a product of functions, bounding the error between the correct and approximate potentials. Recently, researchers have also studied marginal-MAP inference from the perspective of free energy maximization, and proposed message passing approximation algorithms. For example, Jiang, Rai, and Daumé III (2011) propose a hybrid message passing algorithm motivated by a Bethe-like free energy. Liu and Ihler (2011b) provide a general variational framework for marginal MAP, and derive several approximate inference algorithms based on the Bethe and tree-reweighted approximations; the treereweighted approximation provides an upper bound of the optimal energy. In this paper, we explore a two-step approximation methods for marginal-MAP inference, in which we construct an explicit factorized approximation of the marginalized distribution using a form of approximate variable elimination, producing a structured MAP problem that can be solved using a variety of existing methods, such as dual decomposition (Sontag, Globerson, and Jaakkola 2011). We use a novel chain decomposition approach to construct the approximate marginalization, and apply a Hölder inequality (Liu and Ihler 2011a) to obtain bounds on the exact marginalization. This also allows us to interpret our method in terms of an up-