# Densities of Almost Surely Terminating Probabilistic Programs are Differentiable Almost Everywhere

arXiv: https://arxiv.org/abs/2004.03924 · HTML: https://mlsanity.com/paper/2004.03924/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Apr 2020.

## Built on by (most cited first, 6 of 6)

- [Nonparametric Hamiltonian Monte Carlo](https://mlsanity.com/paper/2106.10238/) (2021)
- [ωPAP Spaces: Reasoning Denotationally About Higher-Order, Recursive Probabilistic and Differentiable Programs](https://mlsanity.com/paper/2302.10636/) (2023)
- [Fast and Correct Gradient-Based Optimisation for Probabilistic Programming via Smoothing](https://mlsanity.com/paper/2301.03415/) (2023)
- [Guaranteed Bounds for Posterior Inference in Universal Probabilistic Programming](https://mlsanity.com/paper/2204.02948/) (2022)
- [Nonparametric Involutive Markov Chain Monte Carlo](https://mlsanity.com/paper/2211.01100/) (2022)
- [Diagonalisation SGD: Fast & Convergent SGD for Non-Differentiable Models via Reparameterisation and Smoothing](https://mlsanity.com/paper/2402.11752/) (2024)

## Abstract (arXiv)

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.