Marginal MAP inference involves making MAP predictions in systems defined with latent variables or missing information. It is significantly more difficult than pure marginalization and MAP tasks, for which a large class of efficient and convergent variational algorithms, such as dual decomposition, exist. In this work, we generalize dual decomposition to a generic power sum inference task, which includes marginal MAP, along with pure marginalization and MAP, as special cases. Our method is based on a block coordinate descent algorithm on a new convex decomposition bound, that is guaranteed to converge monotonically, and can be parallelized efficiently. We demonstrate our approach on marginal MAP queries defined on real-world problems from the UAI approximate inference challenge, showing that our framework is faster and more reliable than previous methods.
While our focus is on variational bounds @cite_5 @cite_15 , there are many non-variational approaches for marginal MAP as well. provide upper bounds on marginal MAP by reordering the order in which variables are eliminated, and using exact inference in the reordered join-tree; however, this is exponential in the size of the (unconstrained) treewidth, and can easily become intractable. give an approximation closely related to mini-bucket to bound the marginal MAP; however, unlike (weighted) mini-bucket, these bounds cannot be improved iteratively. The same is true for the algorithm of , which also has a strong dependence on treewidth. Other examples of marginal MAP algorithms include local search [e.g.,][] park2004complexity and Markov chain Monte Carlo methods [e.g.,][] doucet02,yuan04 .
@cite_5 @cite_15
2130467648 2149190985
Marginal MAP problems are known to be very difficult tasks for graphical models and are so far solved exactly by systematic search guided by a join-tree upper bound. In this paper, we develop new AND OR branch and bound algorithms for marginal MAP that use heuristics extracted from weighted mini-buckets enhanced with message-passing updates. We demonstrate the effectiveness of the resulting search algorithms against previous join-tree based approaches, which we also extend to accommodate high induced width models, through extensive empirical evaluations. Our results show not only orders-of-magnitude improvements over the state-of-the-art, but also the ability to solve problem instances well beyond the reach of previous approaches. The MAP (maximum a posteriori hypothesis) problem in Bayesian networks is to find the most likely states of a set of variabls given partial evidence on the complement of that set. Standard structure-based inference methods for finding exact solutions to MAP, such as variable elimination and join-tree algorithms, have complexities that are exponential in the constrained treewidth of the network. A more recent algorithm, proposed by Park and Darwiche, is exponential only in the treewidth and has been shown to handle networks whose constrained treewidth is quite high. In this paper we present a new algorithm for exact MAP that is not necessarily limited in scalability even by the treewidth. This is achieved by leveraging recent advances in compilation of Bayesian networks into arithmetic circuits, which can circumvent treewidth-imposed limits by exploiting the local structure present in the network. Specifically, we implement a branch-and-bound search where the bounds are computed using linear-time operations on the compiled arithmetic circuit. On networks with local structure, we observe orders-of-magnitude improvements over the algorithm of Park and Darwiche. In particular, we are able to efficiently solve many problems where the latter algorithm runs out of memory because of high treewidth. 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 variables 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 marginalization problems. We derive a general dual representation for marginal MAP that naturally integrates the marginalization and maximization operations into a joint variational optimization problem, 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 updates. We also derive a class of convergent algorithms based on proximal point methods, including 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 locally 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. We describe an algorithm for approximate inference in graphical models based on Holder's inequality that provides upper and lower bounds on common summation problems such as computing the partition function or probability of evidence in a graphical model. Our algorithm unifies and extends several existing approaches, including variable elimination techniques such as mini-bucket elimination and variational methods such as tree reweighted belief propagation and conditional entropy decomposition. We show that our method inherits benefits from each approach to provide significantly better bounds on sum-product tasks.