Abstract. We propose a simple approach to combining first-order logic and probabilistic graphical models in a single representation. A Markov logic network (MLN) is a first-order knowledge base with a weight attached to each formula (or clause). Together with a set of constants representing objects in the domain, it specifies a ground Markov network containing one feature for each possible grounding of a first-order formula in the KB, with the corresponding weight. Inference in MLNs is performed by MCMC over the minimal subset of the ground network required for answering the query. Weights are efficiently learned from relational databases by iteratively optimizing a pseudo-likelihood measure. Optionally, additional clauses are learned using inductive logic programming techniques. Experiments with a real-world database and knowledge base in a university domain illustrate the promise of this approach.

Most probabilistic inference algorithms are specified and processed on a propositional level. In the last decade, many proposals for algorithms accepting first-order specifications have been presented, but in the inference stage they still operate on a mostly propositional representation level. [Poole, 2003] presented a method to perform inference directly on the first-order level, but this method is limited to special cases. In this paper we present the first exact inference algorithm that operates directly on a first-order level, and that can be applied to any first-order model (specified in a language that generalizes undirected graphical models). Our experiments show superior performance in comparison with propositional exact inference. 1

Unifying first-order logic and probability is a long-standing goal of AI, and in recent years many representations combining aspects of the two have been proposed. However, inference in them is generally still at the level of propositional logic, creating all ground atoms and formulas and applying standard probabilistic inference methods to the resulting network. Ideally, inference should be lifted as in first-order logic, handling whole sets of indistinguishable objects together, in time independent of their cardinality. Poole (2003) and Braz et al. (2005, 2006) developed a lifted version of the variable elimination algorithm, but it is extremely complex, generally does not scale to realistic domains, and has only been applied to very small artificial problems. In this paper we propose the first lifted version of a scalable probabilistic inference algorithm, belief propagation (loopy or not). Our approach is based on first constructing a lifted network, where each node represents a set of ground atoms that all pass the same messages during belief propagation. We then run belief propagation on this network. We prove the correctness and optimality of our algorithm. Experiments show that it can greatly reduce the cost of inference.

We describe in this paper a system for exact inference with relational Bayesian networks as defined in the publicly available Primula tool. The system is based on compiling propositional instances of relational Bayesian networks into arithmetic circuits and then performing online inference by evaluating and differentiating these circuits in time linear in their size. We report on experimental results showing successful compilation and efficient inference on relational Bayesian networks, whose Primula–generated propositional instances have thousands of variables, and whose jointrees have clusters with hundreds of variables.

Lifted inference algorithms exploit repeated structure in probabilistic models to answer queries efficiently. Previous work such as de Salvo Braz et al.’s first-order variable elimination (FOVE) has focused on the sharing of potentials across interchangeable random variables. In this paper, we also exploit interchangeability within individual potentials by introducing counting formulas, which indicate how many of the random variables in a set have each possible value. We present a new lifted inference algorithm, C-FOVE, that not only handles counting formulas in its input, but also creates counting formulas for use in intermediate potentials. C-FOVE can be described succinctly in terms of six operators, along with heuristics for when to apply them. Because counting formulas capture dependencies among large numbers of variables compactly, C-FOVE achieves asymptotic speed improvements compared to FOVE.

Several real-world applications need to effectively manage and reason about large amounts of data that are inherently uncertain. For instance, pervasive computing applications must constantly reason about volumes of noisy sensory readings for a variety of reasons, including motion prediction and human behavior modeling. Such probabilistic data analyses require sophisticated machine-learning tools that can effectively model the complex spatio/temporal correlation patterns present in uncertain sensory data. Unfortunately, to date, most existing approaches to probabilistic database systems have relied on somewhat simplistic models of uncertainty that can be easily mapped onto existing relational architectures: Probabilistic information is typically associated with individual data tuples, with only limited or no support for effectively capturing and reasoning about complex data correlations. In this paper, we introduce BAYESSTORE, a novel probabilistic data management architecture built on the principle of handling statistical models and probabilistic inference tools as first-class citizens of the database system. Adopting a machine-learning view, BAYESSTORE employs concise statistical relational models to effectively encode the correlation patterns between uncertain data, and promotes probabilistic inference and statistical model manipulation as part of the standard DBMS operator repertoire to support efficient and sound query processing. We present BAYESSTORE’s uncertainty model based on a novel, first-order statistical model, and we redefine traditional query processing operators, to manipulate the data and the probabilistic models of the database in an efficient manner. Finally, we validate our approach, by demonstrating the value of exploiting data correlations during query processing, and by evaluating a number of optimizations which significantly accelerate query processing. 1

There has been a recent surge in work in probabilistic databases, propelled in large part by the huge increase in noisy data sources — from sensor data, experimental data, data from uncurated sources, and many others. There is a growing need for database management systems that can efficiently represent and query such data. In this work, we show how data characteristics can be leveraged to make the query evaluation process more efficient. In particular, we exploit what we refer to as shared correlations where the same uncertainties and correlations occur repeatedly in the data. Shared correlations occur mainly due to two reasons: (1) Uncertainty and correlations usually come from general statistics and rarely vary on a tuple-to-tuple basis; (2) The query evaluation procedure itself tends to re-introduce the same correlations. Prior work has shown that the query evaluation problem on probabilistic databases is equivalent to a probabilistic inference problem on an appropriately constructed probabilistic graphical model (PGM). We leverage this by introducing a new data structure, called the random variable elimination graph (rv-elim graph) that can be built from the PGM obtained from query evaluation. We develop techniques based on bisimulation that can be used to compress the rv-elim graph exploiting the presence of shared correlations in the PGM, the compressed rv-elim graph can then be used to run inference. We validate our methods by evaluating them empirically and show that even with a few shared correlations significant speed-ups are possible.

Propositionalization of a first-order theory followed by satisfiability testing has proved to be a remarkably efficient approach to inference in relational domains such as planning (Kautz & Selman 1996) and verification (Jackson 2000). More recently, weighted satisfiability solvers have been used successfully for MPE inference in statistical relational learners (Singla & Domingos 2005). However, fully instantiating a finite first-order theory requires memory on the order of the number of constants raised to the arity of the clauses, which significantly limits the size of domains it can be applied to. In this paper we propose LazySAT, a variation of the Walk-SAT solver that avoids this blowup by taking advantage of the extreme sparseness that is typical of relational domains (i.e., only a small fraction of ground atoms are true, and most clauses are trivially satisfied). Experiments on entity resolution and planning problems show that LazySAT reduces memory usage by orders of magnitude compared to Walk-SAT, while taking comparable time to run and producing the same solutions.

A major benefit of graphical models is that most knowledge is captured in the model structure. Many models, however, produce inference problems with a lot of symmetries not reflected in the graphical structure and hence not exploitable by efficient inference techniques such as belief propagation (BP). In this paper, we present a new and simple BP algorithm, called counting BP, that exploits such additional symmetries. Starting from a given factor graph, counting BP first constructs a compressed factor graph of clusternodes and clusterfactors, corresponding to sets of nodes and factors that are indistinguishable given the evidence. Then it runs a modified BP algorithm on the compressed graph that is equivalent to running BP on the original factor graph. Our experiments show that counting BP is applicable to a variety of important AI tasks such as (dynamic) relational models and boolean model counting, and that significant efficiency gains are obtainable, often by orders of magnitude. 1

It is often convenient to represent probabilistic models in a first-order fashion, using logical atoms such as random variables parameterized by logical variables. (de Salvo Braz et al., 2005), following (Poole, 2003), give a lifted variable elimination algorithm (FOVE) for computing marginal probabilities from first-order probabilistic models (belief assessment, or BA). FOVE is lifted because it works directly at the first-order level, eliminating all the instantiations of a set of atoms in a single step. Previous work could treat only restricted potential functions. There, atoms ’ instantiations cannot constrain each other: predicates can appear at most once, or logical variables must not interact across atoms. In this paper, we present two contributions. The first one is a significantly more general lifted variable elimination algorithm, FOVE-P, that covers many cases where atoms share logical variables. The second contribution is to use FOVE-P for solving the Most Probable Explanation (MPE) problem, which consists of calculating the most probable assignment of the random variables in a model. We introduce the notion of lifted assignments, a distribution of values to a set of random variables rather than to each individual one. Lifted assignments are cheaper to compute while being as useful as regular assignments over that group. Both contributions advance the theoretical understanding of lifted probabilistic inference.