Densities of Almost Surely Terminating Probabilistic Programs are Differentiable Almost Everywhere

Abstract

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 à la Borgström 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.

On probabilistic termination of functional programs with continuous distributions

“…geo describes a geometric distribution with parameter , 1dRW , a 1-dimensional -biased random walk starting at [44], gr a term analysed in [51] terminating with a probability given as the inverse golden ratio, 3print the natural extension of Ex. 1.1 (2) to 3 recursive calls, bin , a 1-dimensional random walk in one direction [44] and pedestrian a stochastic program modelling a pedestrian taken inspired by [41]. See the appendix for a detailed description of the example terms.…”

“…Even in its current, unoptimized, form our tool is able to compute meaningful lower bounds in a reasonable time. We evaluate our tool on various examples taken from [33,44,51] and [41] (possibly modified to match the CbN evaluation). The computed lower bounds can be found in table 1.…”

“…To resolve the latter we make use of the conditional oracles. For an overview of a similar system of symbolic execution we refer the reader to [41].…”

“…The symbolic execution we present here differs (especially on first glance) from the one used in [41]. In their semantics, a symbolic configuration at all times contains a set of traces that can take this path.…”

“…[54, §4.3.2] and [25]). Moreover [41] have recently shown that AST programs have density (a.k.a. weight) functions that are differentiable almost everywhere.…”