Many representation schemes combining firstorder logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable elimination and belief propagation, neither of which take logical structure into account. We propose the first method that has the full power of both graphical model inference and first-order theorem proving (in finite domains with Herbrand interpretations). We first define probabilistic theorem proving, their generalization, as the problem of computing the probability of a logical formula given the probabilities or weights of a set of formulas. We then show how this can be reduced to the problem of lifted weighted model counting, and develop an efficient algorithm for the latter. We prove the correctness of this algorithm, investigate its properties, and show how it generalizes previous approaches. Experiments show that it greatly outperforms lifted variable elimination when logical structure is present. Finally, we propose an algorithm for approximate probabilistic theorem proving, and show that it can greatly outperform lifted belief propagation. 1

Lifted belief propagation (LBP) can be extremely fast at computing approximate marginal probability distributions over single ground atoms and neighboring ones in the underlying graphical model. It does, however, not prescribe a way to compute joint distributions over pairs, triples or k-tuples of distant ground atoms. In this paper, we present an algorithm, called conditioned LBP, for approximating these distributions. Essentially, we select variables one at a time for conditioning, running lifted belief propagation after each selection. This naive solution, however, recomputes the lifted network in each step from scratch, therefore often canceling the benefits of lifted inference. We show how to avoid this by efficiently computing the lifted network for each conditioning directly from the one already known for the single node marginals. Our experimental results validate that significant efficiency gains are possible and illustrate the potential for second-order parameter estimation of Markov logic networks. 1

Lifted message passing algorithms exploit repeated structure within a given graphical model to answer queries efficiently. Given evidence, they construct a lifted network of supernodes and superpotentials corresponding to sets of nodes and potentials that are indistinguishable given the evidence. Recently, efficient algorithms were presented for updating the structure of an existing lifted network with incremental changes to the evidence. In the inference stage, however, current algorithms need to construct a separate lifted network for each evidence case and run a modified message passing algorithm on each lifted network separately. Consequently, symmetries across the inference tasks are not exploited. In this paper, we present a novel lifted message passing technique that exploits symmetries across multiple evidence cases. The benefits of this multi-evidence lifted inference are shown for several important AI tasks such as computing personalized PageRanks and Kalman filters via multievidence lifted Gaussian belief propagation. 1

Abstract. Markov logic networks (MLNs) have been successfully applied to several challenging problems by taking a “programming language ” approach where a set of formulas is hand-coded and weights are learned from data. Because inference plays an important role in this process, “programming ” with an MLN would be significantly facilitated by speeding up inference. We present a new meta-inference algorithm that exploits the repeated structure frequently present in relational domains to speed up existing inference techniques. Our approach first clusters the query literals and then performs full inference for only one representative from each cluster. The clustering step incurs only a one-time up-front cost when weights are learned over a fixed structure. 1

Coarse-to-fine approaches use sequences of increasingly fine approximations to control the complexity of inference and learning. These techniques are often used in NLP and vision applications. However, no coarse-to-fine inference or learning methods have been developed for general first-order probabilistic domains, where the potential gains are even higher. We present our Coarse-to-Fine Probabilistic Inference (CFPI) framework for general coarse-to-fine inference for first-order probabilistic models, which leverages a given or induced type hierarchy over objects in the domain. Starting by considering the inference problem at the coarsest type level, our approach performs inference at successively finer grains, pruning highand low-probability atoms before refining. CFPI can be applied with any probabilistic inference method and can be used in both propositional and relational domains. CFPI provides theoretical guarantees on the errors incurred, and these guarantees can be tightened when CFPI is applied to specific inference algorithms. We also show how to learn parameters in a coarse-to-fine manner to maximize the efficiency of CFPI. We evaluate CFPI with the lifted belief propagation algorithm on social network link prediction and biomolecular event prediction tasks. These experiments show CFPI can greatly speed up inference without sacrificing accuracy.

Lifting can greatly reduce the cost of inference on firstorder probabilistic models, but constructing the lifted network can itself be quite costly. In addition, the minimal lifted network is often very close in size to the fully propositionalized model; lifted inference yields little or no speedup in these situations. In this paper, we address both these problems. We propose a compact hypercubebased representation for the lifted network, which can greatly reduce the cost of lifted network construction. We also present two methods for approximate lifted network construction, which groups together similar but distinguishable objects and treats them as if they were identical. This can greatly reduce the size of the lifted network as well as the time required for lifted network construction, but potentially at some cost to accuracy. The coarseness of the approximation can be adjusted depending on the accuracy required, and we can bound the resulting error. Experiments on six domains show great efficiency gains with only minor loss in accuracy.

Lifted inference, handling whole sets of indistinguishable objects together, is critical to the effective application of probabilistic relational models to realistic real world tasks. Recently, lifted belief propagation (LBP) has been proposed as an efficient approximate solution of this inference problem. It runs a modified BP on a lifted network where nodes have been grouped together if they have — roughly speaking — identical computation trees, the tree-structured unrolling of the underlying graph rooted at the nodes. In many situations, this purely syntactic criterion is too pessimistic: message errors decay along paths. Intuitively, for a long chain graph with weak edge potentials, distant nodes will send and receive identical messages yet their computation trees are quite different. To overcome this, we propose iLBP, a novel, easy-to-implement, informed LBP approach that interleaves lifting and modified BP iterations. In turn, we can efficiently monitor the true BP messages sent and received in each iteration and group nodes accordingly. As our experiments show, iLBP can yield significantly faster more lifted network while not degrading performance. Above all, we show that iLBP is faster than BP when solving the problem of distributing data to a large network, an important real-world application where BP is faster than uninformed LBP.

Lifted message passing algorithms exploit repeated structure within a given graphical model to answer queries efficiently. Given evidence, they construct a lifted network of supernodes and superpotentials corresponding to sets of nodes and potentials that are indistinguishable given the evidence. Recently, efficient algorithms were presented for updating the structure of an existing lifted network with incremental changes to the evidence. In the inference stage, however, current algorithms need to construct a separate lifted network for each evidence case and run a modified message passing algorithm on each lifted network separately. Consequently, symmetries across the inference tasks are not exploited. In this paper, we present a novel lifted message passing technique that exploits symmetries across multiple evidence cases. The benefits of this multi-evidence lifted inference are shown for several important AI tasks such as computing personalized PageRanks and Kalman filters via multievidence lifted Gaussian belief propagation. 1

Many tasks in AI require representation and manipulation of complex functions. First order decision diagrams (FODD) are a compact knowledge representation expressing functions over relational structures. They represent numerical functions that, when constrained to the Boolean range, use only existential quantification. Previous work has developed a set of operations for composition and for removing redundancies in FODDs, thus keeping them compact, and showed how to successfully employ FODDs for solving large-scale stochastic planning problems through the formalism of relational Markov decision processes (RMDP). In this paper, we introduce several new ideas enhancing the applicability of FODDs. More specifically, we first introduce Generalized FODDs (GFODD) and composition operations for them, generalizing FODDs to arbitrary quantification. Second, we develop a novel approach for reducing (G)FODDs using model checking. This yields – for the first time – a reduction that maximally reduces the diagram for the FODD case and provides a sound reduction procedure for GFODDs. Finally we show how GFODDs can be used in principle to solve RMDPs with arbitrary quantification, and develop a complete solution for the case where the reward function is specified using an arbitrary number of existential quantifiers followed by an arbitrary number of universal quantifiers. Keywords:

Abstract. Lifted message passing approaches can be extremely fast at computing approximate marginal probability distributions over single variables and neighboring ones in the underlying graphical model. They do, however, not prescribe a way to solve more complex inference tasks such as computing joint marginals for k-tuples of distant random variables or satisfying assignments of CNFs. A popular solution in these cases is the idea of turning the complex inference task into a sequence of simpler ones by selecting and clamping variables one at a time and running lifted message passing again after each selection. This naive solution, however, recomputes the lifted network in each step from scratch, therefore often canceling the benefits of lifted inference. We show how to avoid this by efficiently computing the lifted network for each conditioning directly from the one already known for the single node marginals. Our experiments show that significant efficiency gains are possible for lifted message passing guided decimation for SAT and sampling.