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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 \u00e0 la Borgstr\u00f6m et al., and study the associated weight (commonly referred to as the density) function and value function on the set of possible execution traces.<\/jats:p><jats:p>Our main result is that almost surely terminating SPCF programs, generated from a set of primitive functions (e.g.\u00a0the 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<jats:italic>deterministic<\/jats:italic>(S)PCF programs with real parameters denote functions that are almost everywhere differentiable.<\/jats:p><jats:p>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.<\/jats:p>","DOI":"10.1007\/978-3-030-72019-3_16","type":"book-chapter","created":{"date-parts":[[2021,3,22]],"date-time":"2021-03-22T18:03:10Z","timestamp":1616436190000},"page":"432-461","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Densities of Almost Surely Terminating Probabilistic Programs are Differentiable Almost Everywhere"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6512-2864","authenticated-orcid":false,"given":"Carol","family":"Mak","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7509-680X","authenticated-orcid":false,"given":"C.-H. 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