Abstract
AbstractWe give two geometry of interaction models for a typed λ-calculus with recursion endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The models are based on the category of measurable spaces and partial measurable functions, and the category of measurable spaces and s-finite kernels, respectively. The former is proved adequate with respect to both a distribution-based and a sampling-based operational semantics, while the latter is proved adequate with respect to a sampling-based operational semantics.
Highlights
Randomisation provides the most efficient algorithmic solutions, at least concretely, in many different contexts
The distribution-based operational semantics of PCFSS is a family of binary relations {⇒n}n∈N between closed terms of type Real and measures on R inductively defined by the evaluation rules in Figure 3, where the evaluation rule for score is inspired from the one in Staton (2017)
L!, which sends an object A ∈ L! to State State⊗!A. This use of the state monad is motivated by sampling-based operational semantics: we can regard PCFSS as a call-by-value λ-calculus with global states consisting of pairs of a non-negative real number and a finite sequence of real numbers, and we can regard score and sample as effectful operations interacting with those states
Summary
Randomisation provides the most efficient algorithmic solutions, at least concretely, in many different contexts. Starting from a monad together with some algebraic effects, they gave an adequate GoI model for such a calculus, which is applicable to a wide range of algebraic effects In principle, their recipe could be applicable to PCFSS, since samplingbased operational semantics enables us to see scoring and sampling as algebraic effects acting on global states. For more detail on categorical aspect of our semantics, see Dal Lago and Hoshino (2019a) The use of such compact closed categories (or, more generally, of traced monoidal categories) is the way GoI models higher-order functions. This paper is a revised and extended version of our conference paper (Dal Lago and Hoshino 2019b)
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