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

Probabilistic Programming Semantics for Name Generation

We make a formal analogy between random sampling and fresh name generation. We show that quasi-Borel spaces, a model for probabilistic programming, can soundly interpret Stark's $\nu$-calculus, a calculus for name generation. Moreover, we prove that this semantics is fully abstract up to first-order types. This is surprising for an 'off-the-shelf' model, and requires a novel analysis of probability distributions on function spaces. Our tools are diverse and include descriptive set theory and normal forms for the $\nu$-calculus.Comment: 29 pages, 1 figure; to be published in POPL 202

Trace types and denotational semantics for sound programmable inference in probabilistic languages

Modern probabilistic programming languages aim to formalize and automate key aspects of probabilistic modeling and inference. Many languages provide constructs for programmable inference that enable developers to improve inference speed and accuracy by tailoring an algorithm for use with a particular model or dataset. Unfortunately, it is easy to use these constructs to write unsound programs that appear to run correctly but produce incorrect results. To address this problem, we present a denotational semantics for programmable inference in higher-order probabilistic programming languages, along with a type system that ensures that well-typed inference programs are sound by construction. A central insight is that the type of a probabilistic expression can track the space of its possible execution traces, not just the type of value that it returns, as these traces are often the objects that inference algorithms manipulate. We use our semantics and type system to establish soundness properties of custom inference programs that use constructs for variational, sequential Monte Carlo, importance sampling, and Markov chain Monte Carlo inference

SPPL: Probabilistic Programming with Fast Exact Symbolic Inference

We present the Sum-Product Probabilistic Language (SPPL), a new probabilistic programming language that automatically delivers exact solutions to a broad range of probabilistic inference queries. SPPL translates probabilistic programs into sum-product expressions, a new symbolic representation and associated semantic domain that extends standard sum-product networks to support mixed-type distributions, numeric transformations, logical formulas, and pointwise and set-valued constraints. We formalize SPPL via a novel translation strategy from probabilistic programs to sum-product expressions and give sound exact algorithms for conditioning on and computing probabilities of events. SPPL imposes a collection of restrictions on probabilistic programs to ensure they can be translated into sum-product expressions, which allow the system to leverage new techniques for improving the scalability of translation and inference by automatically exploiting probabilistic structure. We implement a prototype of SPPL with a modular architecture and evaluate it on benchmarks the system targets, showing that it obtains up to 3500x speedups over state-of-the-art symbolic systems on tasks such as verifying the fairness of decision tree classifiers, smoothing hidden Markov models, conditioning transformed random variables, and computing rare event probabilities