Abstract

We give a geometry of interaction model for a typed lambda-calculus endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The model is based on the category of measurable spaces and partial measurable functions, and is proved adequate with respect to both a distribution-based and a sampling based operational semantics.

Highlights

  • Randomisation provides the most efficient algorithmic solutions, at least concretely, in many different contexts

  • Giving a satisfactory denotational semantics to higher-order functional languages is already problematic in presence of probabilistic choice [6, 14], and becomes even more challenging when continuous distributions and scoring are present

  • We introduced a denotational semantics for PCFSS, a higher-order functional language with sampling from a uniform continuous distribution and scoring

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Summary

Introduction

Randomisation provides the most efficient algorithmic solutions, at least concretely, in many different contexts. Giving a satisfactory denotational semantics to higher-order functional languages is already problematic in presence of probabilistic choice [6, 14], and becomes even more challenging when continuous distributions and scoring are present. Quasi-Borel spaces [15] have been proposed as a way to give semantics to calculi with all these features, and only very recently [16] this framework has been shown to be adaptable to a fully-fledged calculus for probabilistic programming, in which continuous distributions and soft-conditioning are present. This paper’s contributions, beside the model’s definition, are two adequacy results which precisely relate our GoI model to the operational semantics, as expressed (following [28]), in both the distribution and sampling styles. As a corollary of our adequacy results, we show that integrating over the sampling-based operational semantics, one obtains precisely the distribution-based operational semantics

Turning Measurable Spaces into a GoI Model
Outline
Measure-Theoretic Preliminaries
Syntax and Type System
Distribution-Based Operational Semantics
Sampling-Based Operational Semantics
Towards Mealy Machine Semantics
Moggi’s Translation
Girard Translation
The Third Step
From Proof Structures to Mealy Machines
Mealy Machines and their Compositions
Behavioural Equivalence
Constructions on Mealy Machines
Composition
Monoidal Products
Symmetry
Real Numbers
Measurable Functions
Conditional Branching
A State Monad
5.4.10 Scoring
Diagrammatic Reasoning
The Category of Partial Measurable Functions
The Category of Mealy Machines
Mealy Machine Semantics for PCFSS
Proof of Adequacy Theorems
Approximation Lemma
10 How About S-Finite Kernels?
10.1 S-finite Kernels
10.2 Probabilistic Mealy Machine
10.3 Behavioral Equivalence
10.4 Construction of probabilistic Mealy Machines
10.4.2 Monoidal Products
10.4.3 A Modal Operator
10.4.4 Diagrammatic Reasoning on Probabilistic Mealy Machines
10.4.5 A State Monad
10.4.7 Sampling
11 Probabilistic Mealy Machine Semantics for PCFSS
11.1.1 Observation
11.1.3 Induction step on recursion
11.1.4 Commutativity Modulo Observational Equivalence
12 Conclusion
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