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.

Building on the observation that reverse-mode automatic differentiation (AD) -a generalisation of backpropagation -can naturally be expressed as pullbacks of differential 1forms, we design a simple higher-order programming language with a first-class differential operator, and present a reduction strategy which exactly simulates reverse-mode AD. We justify our reduction strategy by interpreting our language in any differential λ-category that satisfies the Hahn-Banach Separation Theorem, and show that the reduction strategy precisely captures reverse-mode AD in a truly higherorder setting.

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