We define a new games model of Probabilistic PCF (PPCF) by enriching thin concurrent games with symmetry, recently introduced by Castellan et al, with probability. This model supports two interpretations of PPCF, one sequential and one parallel. We make the case for this model by exploiting the causal structure of probabilistic concurrent strategies. First, we show that the strategies obtained from PPCF programs have a deadlock-free interaction, and therefore deduce that there is an interpretation-preserving functor from our games to the probabilistic relational model recently proved fully abstract by Ehrhard et al. It follows that our model is intensionally fully abstract. Finally, we propose a definition of probabilistic innocence and prove a finite definability result, leading to a second (independent) proof of full abstraction.
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.
We define a new denotational semantics for a first-order probabilistic programming language in terms of probabilistic event structures. This semantics is intensional, meaning that the interpretation of a program contains information about its behaviour throughout execution, rather than a simple distribution on return values. In particular, occurrences of sampling and conditioning are recorded as explicit events, partially ordered according to the data dependencies between the corresponding statements in the program.This interpretation is adequate: we show that the usual measuretheoretic semantics of a program can be recovered from its event structure representation. Moreover it can be leveraged for MCMC inference: we prove correct a version of single-site Metropolis-Hastings with incremental recomputation, in which the proposal kernel takes into account the semantic information in order to avoid performing some of the redundant sampling.
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