First-order Probabilistic Models Research Articles

Higher Order Bayesian Networks, Exactly

Bayesian networks are graphical first-order probabilistic models that allow for a compact representation of large probability distributions, and for efficient inference, both exact and approximate. We introduce a higher-order programming language, in the idealized form of a lambda-calculus, which we prove sound and complete w.r.t. Bayesian networks: each Bayesian network can be encoded as a term, and conversely each (possibly higher-order and recursive) program of ground type compiles into a Bayesian network. The language allows for the specification of recursive probability models and hierarchical structures. Moreover, we provide a compositional and cost-aware semantics which is based on factors, the standard mathematical tool used in Bayesian inference. Our results rely on advanced techniques rooted into linear logic, intersection types, rewriting theory, and Girard's geometry of interaction, which are here combined in a novel way.

Counting-MLNs: Learning Relational Structure for Decision Making

Many first-order probabilistic models can be represented much more compactly using aggregation operations such as counting. While traditional statistical relational representations share factors across sets of interchangeable random variables, representations that explicitly model aggregations also exploit interchangeability of random variables within factors. This is especially useful in decision making settings, where an agent might need to reason about counts of the different types of objects it interacts with. Previous work on counting formulas in statistical relational representations has mostly focused on the problem of exact inference on an existing model. The problem of learning such models is largely unexplored. In this paper, we introduce Counting Markov Logic Networks (C-MLNs), an extension of Markov logic networks that can compactly represent complex counting formulas. We present a structure learning algorithm for C-MLNs; we apply this algorithm to the novel problem of generalizing natural language instructions, and to relational reinforcement learning in the Crossblock domain, in which standard MLN learning algorithms fail to find any useful structure. The C-MLN policies learned from natural language instructions are compact and intuitive, and, despite requiring no instructions on test games, win 20% more Crossblock games than a state-of-the-art algorithm for following natural language instructions.

Coarse-to-Fine Inference and Learning for First-Order Probabilistic Models

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 high- and 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.

Learning first-order probabilistic models with combining rules

Many real-world domains exhibit rich relational structure and stochasticity and motivate the development of models that combine predicate logic with probabilities. These models describe probabilistic influences between attributes of objects that are related to each other through known domain relationships. To keep these models succinct, each such influence is considered independent of others, which is called the assumption of "independence of causal influences" (ICI). In this paper, we describe a language that consists of quantified conditional influence statements and captures most relational probabilistic models based on directed graphs. The influences due to different statements are combined using a set of combining rules such as Noisy-OR. We motivate and introduce multi-level combining rules, where the lower level rules combine the influences due to different ground instances of the same statement, and the upper level rules combine the influences due to different statements. We present algorithms and empirical results for parameter learning in the presence of such combining rules. Specifically, we derive and implement algorithms based on gradient descent and expectation maximization for different combining rules and evaluate them on synthetic data and on a real-world task. The results demonstrate that the algorithms are able to learn both the conditional probability distributions of the influence statements and the parameters of the combining rules.