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We present SGDPLL(T), an algorithm that solves (among many other problems) probabilistic inference modulo theories, that is, inference problems over probabilistic models defined via a logic theory provided as a parameter (currently, propositional, equalities on discrete sorts, and inequalities, more specifically difference arithmetic, on bounded integers). While many solutions to probabilistic inference over logic representations have been proposed, SGDPLL(T) is simultaneously (1) lifted, (2) exact and (3) modulo theories, that is, parameterized by a background logic theory. This offers a foundation for extending it to rich logic languages such as data structures and relational data. By lifted, we mean algorithms with constant complexity in the domain size (the number of values that variables can take). We also detail a solver for summations with difference arithmetic and show experimental results from a scenario in which SGDPLL(T) is much faster than a state-of-the-art probabilistic solver., Submitted to StarAI-16 workshop as closely revised version of IJCAI-16 paper

Probabilistic inference algorithms for finding the most probable explanation, the maximum aposteriori hypothesis, and the maximum expected utility and for updating belief are reformulated as an elimination--type algorithm called bucket elimination. This emphasizes the principle common to many of the algorithms appearing in that literature and clarifies their relationship to nonserial dynamic programming algorithms. We also present a general way of combining conditioning and elimination within this framework. Bounds on complexity are given for all the algorithms as a function of the problem's structure., Appears in Proceedings of the Twelfth Conference on Uncertainty in Artificial Intelligence (UAI1996)

Many algorithms for processing probabilistic networks are dependent on the topological properties of the problem's structure. Such algorithms (e.g., clustering, conditioning) are effective only if the problem has a sparse graph captured by parameters such as tree width and cycle-cut set size. In this paper we initiate a study to determine the potential of structure-based algorithms in real-life applications. We analyze empirically the structural properties of problems coming from the circuit diagnosis domain. Specifically, we locate those properties that capture the effectiveness of clustering and conditioning as well as of a family of conditioning+clustering algorithms designed to gradually trade space for time. We perform our analysis on 11 benchmark circuits widely used in the testing community. We also report on the effect of ordering heuristics on tree-clustering and show that, on our benchmarks, the well-known max-cardinality ordering is substantially inferior to an ordering called min-degree., Appears in Proceedings of the Twelfth Conference on Uncertainty in Artificial Intelligence (UAI1996)

We show that the d -separation criterion constitutes a valid test for conditional independence relationships that are induced by feedback systems involving discrete variables., Appears in Proceedings of the Twelfth Conference on Uncertainty in Artificial Intelligence (UAI1996)

Graphical models are one of the most prominent frameworks to model complex systems and efficiently query them. Their underlying algebraic properties are captured by a valuation structure that, most usually, is a semiring. Depending on the semiring of choice, we can capture probabilistic models, constraint networks, cost networks, etc. In this paper we address the partitioning problem which occurs in many approximation techniques such as mini-bucket elimination and join-graph propagation algorithms. Roghly speaking, subject to complexity bounds, the algorithm needs to find a partition of a set of factors such that best approximates the whole set. While this problem has been addressed in the past in a particular case, we present here a general description. Furthermore, we also propose a general partitioning scheme. Our proposal is general in the sense that it is presented in terms of a generic semiring with the only additional requirements of a division operation and a refinement of its order. The proposed algorithm instantiates to the particular task of computing the probability of evidence, but also applies directly to other important reasoning tasks. We demonstrate its good empirical behaviour on the problem of computing the most probable explanation.

The paper presents an iterative version of join-tree clustering that applies the message passing of join-tree clustering algorithm to join-graphs rather than to join-trees, iteratively. It is inspired by the success of Pearl's belief propagation algorithm as an iterative approximation scheme on one hand, and by a recently introduced mini-clustering i. success as an anytime approximation method, on the other. The proposed Iterative Join-graph Propagation IJGP belongs to the class of generalized belief propagation methods, recently proposed using analogy with algorithms in statistical physics. Empirical evaluation of this approach on a number of problem classes demonstrates that even the most time-efficient variant is almost always superior to IBP and MC i, and is sometimes more accurate by as much as several orders of magnitude., Appears in Proceedings of the Eighteenth Conference on Uncertainty in Artificial Intelligence (UAI2002)

The paper continues the study of partitioning based inference of heuristics for search in the context of solving the Most Probable Explanation task in Bayesian Networks. We compare two systematic Branch and Bound search algorithms, BBBT (for which the heuristic information is constructed during search and allows dynamic variable/value ordering) and its predecessor BBMB (for which the heuristic information is pre-compiled), against a number of popular local search algorithms for the MPE problem. We show empirically that, when viewed as approximation schemes, BBBT/BBMB are superior to all of these best known SLS algorithms, especially when the domain sizes increase beyond 2. This is in contrast with the performance of SLS vs. systematic search on CSP/SAT problems, where SLS often significantly outperforms systematic algorithms. As far as we know, BBBT/BBMB are currently the best performing algorithms for solving the MPE task., Appears in Proceedings of the Nineteenth Conference on Uncertainty in Artificial Intelligence (UAI2003)

We study the problem of complexity estimation in the context of parallelizing an advanced Branch and Bound-type algorithm over graphical models. The algorithm's pruning power makes load balancing, one crucial element of every distributed system, very challenging. We propose using a statistical regression model to identify and tackle disproportionally complex parallel subproblems, the cause of load imbalance, ahead of time. The proposed model is evaluated and analyzed on various levels and shown to yield robust predictions. We then demonstrate its effectiveness for load balancing in practice., Appears in Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence (UAI2012)

This is the Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence, which was held in Cambridge, MA, July 13 - 16 2006.

The paper introduces mixed networks, a new framework for expressing and reasoning with probabilistic and deterministic information. The framework combines belief networks with constraint networks, defining the semantics and graphical representation. We also introduce the AND/OR search space for graphical models, and develop a new linear space search algorithm. This provides the basis for understanding the benefits of processing the constraint information separately, resulting in the pruning of the search space. When the constraint part is tractable or has a small number of solutions, using the mixed representation can be exponentially more effective than using pure belief networks which odel constraints as conditional probability tables., Appears in Proceedings of the Twentieth Conference on Uncertainty in Artificial Intelligence (UAI2004)

In this paper, we consider Hybrid Mixed Networks (HMN) which are Hybrid Bayesian Networks that allow discrete deterministic information to be modeled explicitly in the form of constraints. We present two approximate inference algorithms for HMNs that integrate and adjust well known algorithmic principles such as Generalized Belief Propagation, Rao-Blackwellised Importance Sampling and Constraint Propagation to address the complexity of modeling and reasoning in HMNs. We demonstrate the performance of our approximate inference algorithms on randomly generated HMNs., Appears in Proceedings of the Twenty-First Conference on Uncertainty in Artificial Intelligence (UAI2005)

The paper analyzes theoretically and empirically the performance of likelihood weighting (LW) on a subset of nodes in Bayesian networks. The proposed scheme requires fewer samples to converge due to reduction in sampling variance. The method exploits the structure of the network to bound the complexity of exact inference used to compute sampling distributions, similar to Gibbs cutset sampling. Yet, the extension of the previosly proposed cutset sampling principles to likelihood weighting is non-trivial due to differences in the sampling processes of Gibbs sampler and LW. We demonstrate empirically that likelihood weighting on a cutset (LWLC) is effective time-wise and has a lower rejection rate than LW when applied to networks with many deterministic probabilities. Finally, we show that the performance of likelihood weighting on a cutset can be improved further by caching computed sampling distributions and, consequently, learning 'zeros' of the target distribution., Appears in Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence (UAI2006)

Computing the probability of evidence even with known error bounds is NP-hard. In this paper we address this hard problem by settling on an easier problem. We propose an approximation which provides high confidence lower bounds on probability of evidence but does not have any guarantees in terms of relative or absolute error. Our proposed approximation is a randomized importance sampling scheme that uses the Markov inequality. However, a straight-forward application of the Markov inequality may lead to poor lower bounds. We therefore propose several heuristic measures to improve its performance in practice. Empirical evaluation of our scheme with state-of- the-art lower bounding schemes reveals the promise of our approach., Appears in Proceedings of the Twenty-Third Conference on Uncertainty in Artificial Intelligence (UAI2007)

Mini-Bucket Elimination (MBE) is a well-known approximation algorithm deriving lower and upper bounds on quantities of interest over graphical models. It relies on a procedure that partitions a set of functions, called bucket, into smaller subsets, called mini-buckets. The method has been used with a single partitioning heuristic throughout, so the impact of the partitioning algorithm on the quality of the generated bound has never been investigated. This paper addresses this issue by presenting a framework within which partitioning strategies can be described, analyzed and compared. We derive a new class of partitioning heuristics from first-principles geared for likelihood queries, demonstrate their impact on a number of benchmarks for probabilistic reasoning and show that the results are competitive (often superior) to state-ofthe-art bounding schemes.