
Universal Semantics for the Stochastic LambdaCalculus
We define sound and adequate denotational and operational semantics for ...
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Semantics of higherorder probabilistic programs with conditioning
We present a denotational semantics for higherorder probabilistic progr...
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Discontinuous Hamiltonian Monte Carlo for Probabilistic Programs
Hamiltonian Monte Carlo (HMC) is the dominant statistical inference algo...
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Formal verification of higherorder probabilistic programs
Probabilistic programming provides a convenient lingua franca for writin...
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A Semantics for Probabilistic ControlFlow Graphs
This article develops a novel operational semantics for probabilistic co...
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Probabilistic Programs with Stochastic Conditioning
We propose to distinguish between deterministic conditioning, that is, c...
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Measurable Cones and Stable, Measurable Functions
We define a notion of stable and measurable map between cones endowed wi...
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Densities of almostsurely terminating probabilistic programs are differentiable almost everywhere
We study the differential properties of higherorder 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 callbyvalue PCF with real numbers, and constructs for sampling and conditioning. We give SPCF a samplingstyle 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 almostsurely 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 almosteverywhere differentiable. We use a stochastic form of symbolic execution to reason about almosteverywhere differentiability. A byproduct of this work is that almostsurely terminating deterministic (S)PCF programs with real parameters denote functions that are almosteverywhere differentiable. Our result is of practical interest, as almosteverywhere differentiability of the density function is required to hold for the correctness of major gradientbased inference algorithms.
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