Does a Program Yield the Right Distribution?

Abstract

We study discrete probabilistic programs with potentially unbounded looping behaviors over an infinite state space. We present, to the best of our knowledge, the first decidability result for the problem of determining whether such a program generates exactly a specified distribution over its outputs (provided the program terminates almost-surely). The class of distributions that can be specified in our formalism consists of standard distributions (geometric, uniform, etc.) and finite convolutions thereof. Our method relies on representing these (possibly infinite-support) distributions as probability generating functions which admit effective arithmetic operations. We have automated our techniques in a tool called $$\textsc {Prodigy}$$ P R O D I G Y , which supports automatic invariance checking, compositional reasoning of nested loops, and efficient queries to the output distribution, as demonstrated by experiments.

Inference of Probabilistic Programs with Moment-Matching Gaussian Mixtures

“…All the mentioned methods use a pdf representation of the distributions. More recently, representations using generating functions have been investigated, but only for discrete distributions [Chen et al 2022]. Finally, some approaches use moment-based invariants [Barthe et al 2016;Bartocci et al 2020;Chakarov and Sankaranarayanan 2014;Katoen et al 2010;Moosbrugger et al 2022].…”

Exact Probabilistic Inference Using Generating Functions

“…This representation comes with several benefits: (a) it naturally encodes common, infinite-support distributions (and variations thereof) like the geometric or Poisson distribution in compact, closed-form representations; (b) it allows for compositional reasoning and, in particular, in contrast to representations in terms of density or mass functions, the effective computation of (high-order) moments; (c) tail bounds, concentration bounds, and other properties of interest can be extracted with relative ease from a PGF; and (d) expressions containing parameters, both for probabilities and for assigning new values to program variables, are naturally supported. Some successfully implemented ideas based on PGFs, e.g., for deciding probabilistic equivalence and for proving ⋆ Extended abstract accepted by LAFI 2023 -the Languages for Inference workshop co-located with POPL 2023. non-almost-sure termination, are presented in [5,13], which address especially the aforementioned challenges (1) and (2) for exact probabilistic inference without conditioning.…”

“…Our current research aims to extend the PGF approach towards exact inference for probabilistic programs with conditioning -thus addressing challenges (1), (2), and (3) -and to push the limits of automation as far as possible. To this end, we are in the process of developing an exact, symbolic inference engine based on the open-source, PGF-based tool PRODIGY [5]. We illustrate below its current capability to cater for conditioning via two examples.…”

“…2 cannot, as (λ)PSI do not support programs with unbounded looping behaviors. However, given a suitable invariant as described in [5], PRODIGY is able to reason about the posterior distribution of Prog. 2 in an automated fashion using the second-order PGF (SOP) technique [5]: the resulting posterior distribution for any input with c = 0 is 3•c (4−c 2 ) which encodes precisely a closed-form solution for the generating function…”

“…This is precisely our current research focus. One promising idea is to develop a non-trivial syntactic restriction of the programming language, where the more advanced SOP technique [5] can be generalized to address conditioning inside loops.…”