Probabilistic Inference Modulo Theories

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.Comment: Submitted to StarAI-16 workshop as closely revised version of IJCAI-16 pape

Closed-Form Gibbs Sampling for Graphical Models with Algebraic Constraints

Probabilistic inference in many real-world problems requires graphical models with deterministic algebraic constraints between random variables (e.g., Newtonian mechanics, Pascal’s law, Ohm’s law) that are known to be problematic for many inference methods such as Monte Carlo sampling. Fortunately, when such constraintsare invertible, the model can be collapsed and the constraints eliminated through the well-known Jacobian-based change of variables. As our first contributionin this work, we show that a much broader classof algebraic constraints can be collapsed by leveraging the properties of a Dirac delta model of deterministic constraints. Unfortunately, the collapsing processcan lead to highly piecewise densities that pose challenges for existing probabilistic inference tools. Thus,our second contribution to address these challenges is to present a variation of Gibbs sampling that efficiently samples from these piecewise densities. The key insight to achieve this is to introduce a class of functions that (1) is sufficiently rich to approximate arbitrary models up to arbitrary precision, (2) is closed under dimension reduction (collapsing) for models with (non)linear algebraic constraints and (3) always permits one analytical integral sufficient to automatically derive closed-form conditionals for Gibbs sampling. Experiments demonstrate the proposed sampler converges at least an order of magnitude faster than existing Monte Carlo samplers