Densities of almost-surely terminating probabilistic programs are differentiable almost everywhere

We study the differential properties of higher-order statistical probabilistic programs with recursion and conditioning. Our starting point is an open problem posed by Hongseok Yang: what class of statistical probabilistic programs have densities that are differentiable almost everywhere? To formalise the problem, we consider Statistical PCF (SPCF), an extension of call-by-value PCF with real numbers, and constructs for sampling and conditioning. We give SPCF a sampling-style operational semantics a la Borgstrom et al., and study the associated weight (commonly referred to as the density) function and value function on the set of possible execution traces. Our main result is that almost-surely terminating SPCF programs, generated from a set of primitive functions (e.g. the set of analytic functions) satisfying mild closure properties, have weight and value functions that are almost-everywhere differentiable. We use a stochastic form of symbolic execution to reason about almost-everywhere differentiability. A by-product of this work is that almost-surely terminating deterministic (S)PCF programs with real parameters denote functions that are almost-everywhere differentiable. Our result is of practical interest, as almost-everywhere differentiability of the density function is required to hold for the correctness of major gradient-based inference algorithms

Template-Based Static Posterior Inference for Bayesian Probabilistic Programming

In Bayesian probabilistic programming, a central problem is to estimate the normalised posterior distribution (NPD) of a probabilistic program with conditioning. Prominent approximate approaches to address this problem include Markov chain Monte Carlo and variational inference, but neither can generate guaranteed outcomes within limited time. Moreover, most existing formal approaches that perform exact inference for NPD are restricted to programs with closed-form solutions or bounded loops/recursion. A recent work (Beutner et al., PLDI 2022) derived guaranteed bounds for NPD over programs with unbounded recursion. However, as this approach requires recursion unrolling, it suffers from the path explosion problem. Furthermore, previous approaches do not consider score-recursive probabilistic programs that allow score statements inside loops, which is non-trivial and requires careful treatment to ensure the integrability of the normalising constant in NPD. In this work, we propose a novel automated approach to derive bounds for NPD via polynomial templates. Our approach can handle probabilistic programs with unbounded while loops and continuous distributions with infinite supports. The novelties in our approach are three-fold: First, we use polynomial templates to circumvent the path explosion problem from recursion unrolling; Second, we derive a novel multiplicative variant of Optional Stopping Theorem that addresses the integrability issue in score-recursive programs; Third, to increase the accuracy of the derived bounds via polynomial templates, we propose a novel technique of truncation that truncates a program into a bounded range of program values. Experiments over a wide range of benchmarks demonstrate that our approach is time-efficient and can derive bounds for NPD that are comparable with (or tighter than) the recursion-unrolling approach (Beutner et al., PLDI 2022)

Barrier Frank-Wolfe for Marginal Inference

We introduce a globally-convergent algorithm for optimizing the tree-reweighted (TRW) variational objective over the marginal polytope. The algorithm is based on the conditional gradient method (Frank-Wolfe) and moves pseudomarginals within the marginal polytope through repeated maximum a posteriori (MAP) calls. This modular structure enables us to leverage black-box MAP solvers (both exact and approximate) for variational inference, and obtains more accurate results than tree-reweighted algorithms that optimize over the local consistency relaxation. Theoretically, we bound the sub-optimality for the proposed algorithm despite the TRW objective having unbounded gradients at the boundary of the marginal polytope. Empirically, we demonstrate the increased quality of results found by tightening the relaxation over the marginal polytope as well as the spanning tree polytope on synthetic and real-world instances.Comment: 25 pages, 12 figures, To appear in Neural Information Processing Systems (NIPS) 2015, Corrected reference and cleaned up bibliograph

Automatic Backward Filtering Forward Guiding for Markov processes and graphical models

We incorporate discrete and continuous time Markov processes as building blocks into probabilistic graphical models with latent and observed variables. We introduce the automatic Backward Filtering Forward Guiding (BFFG) paradigm (Mider et al., 2020) for programmable inference on latent states and model parameters. Our starting point is a generative model, a forward description of the probabilistic process dynamics. We backpropagate the information provided by observations through the model to transform the generative (forward) model into a pre-conditional model guided by the data. It approximates the actual conditional model with known likelihood-ratio between the two. The backward filter and the forward change of measure are suitable to be incorporated into a probabilistic programming context because they can be formulated as a set of transformation rules. The guided generative model can be incorporated in different approaches to efficiently sample latent states and parameters conditional on observations. We show applicability in a variety of settings, including Markov chains with discrete state space, interacting particle systems, state space models, branching diffusions and Gamma processes