2020/04/30 by Carol Mak, C. -H. Luke Ong, C.-H. Luke Ong +2
Computer Science · Decision Sciences · #Almost everywhere #Bayesian Modeling and Causal Inference #Class (philosophy) #Correctness #Differentiable function #Function (biology) #Logic, Reasoning, and Knowledge #Probabilistic logic #Recursion (computer science) #Risk and Portfolio Optimization #Set (abstract data type) #cs.LG #cs.LO #cs.PL
paper · pdf · doi:10.1007/978-3-030-72019-3_16
openalex created_date 2020/04/17 · openalex publication_date 2021/01/01 · arxiv created 2021/06/21 · arxiv updated 2021/06/22 · openalex updated_date 2026/08/05
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.