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Deriving the Giry algebras on standard Borel spaces using $\mathbb{R}_{\infty}$-generalized points

Kirk Sturtz

TL;DR

The paper proves that the algebras of the Giry monad on standard spaces are precisely the objects of the full subcategory $\mathbf{Std}_{Cvx}$, i.e., standard measurable spaces equipped with a convex structure satisfying a fullness property. The central technique is to view probability measures as $\mathbb{R}_{\infty}$-valued functionals and to use the single object $\mathbb{R}_{\infty}$ as a codense generator, enabling an adjoint factorization of the Giry monad: $\mathbf{Std} \xrightarrow{\hat{\mathcal{G}}} \mathbf{Std}_{Cvx} \xrightarrow{\mathcal{U}_{Cvx}} \mathbf{Std}$. The authors construct $\,\mathcal{G}$-algebras on all objects of $\mathbf{Std}_{Cvx}$ via the expectation maps $\mathbb{E}_{\bullet}(id_A)$ and show conversely that every $\mathcal{G}$-algebra must lie in $\mathbf{Std}_{Cvx}$, culminating in the equality $\mathbf{Std}_{Cvx}=\mathbf{Alg}_{\mathcal{G}}$. This yields a concrete, algebraic, and expectation-centric understanding of probabilistic computation on standard spaces.

Abstract

The Giry monad on the category of measurable spaces restricts to the full subcategory of standard Borel spaces, $\mathbf{Std}$, which we show is amenable to analysis. $\mathbf{Std}$ contains the space $\mathbb{R}_{\infty}$ which is the one-point compactification of the real numbers. By viewing probability measures $P \in \mathcal{G}(A)$ as functionals operating on measurable functions $A \rightarrow \mathbb{R}_{\infty}$, and taking the restriction of those functionals to operate on affine measurable functions we show that $A \cong Hom_{\mathbb{R}_{\infty}^{\mathbb{R}_{\infty}}}(\mathbb{R}_{\infty}^A|,\mathbb{R}_{\infty})$ for all object $A$ lying in the subcategory $\mathbf{Std}_{Cvx}$ of $\mathbf{Std}$. The objects of $\mathbf{Std}_{Cvx}$ are standard spaces with a convex space structure which satisfies the generic ``fullness property''. The morphisms of the category $\mathbf{Std}_{Cvx}$ are affine measurable functions. The isomorphism is equivalent to the statement that the full subcategory of $\mathbf{Std}_{Cvx}$ consisting of the single object $\mathbb{R}_{\infty}$ is codense in $\mathbf{Std}_{Cvx}$ which allows us to easily construct the $\mathcal{G}$-algebras of objects in $\mathbf{Std}_{Cvx}$. This permits an adjoint factorization of the Giry monad as the composite of $\mathbf{Std} \xrightarrow{\hat{\mathcal{G}}} \mathbf{Std}_{Cvx}$, which is the Giry monad functor viewed as a functor into $\mathbf{Std}_{Cvx}$, and the partial forgetful functor $\mathbf{Std}_{Cvx} \xrightarrow{\mathcal{U}_{Cvx}} \mathbf{Std}$ which forgets the convex space structure. We prove that the category $\mathbf{Std}_{Cvx}$ is the category of algebras of the $\mathcal{G}$-monad.

Deriving the Giry algebras on standard Borel spaces using $\mathbb{R}_{\infty}$-generalized points

TL;DR

The paper proves that the algebras of the Giry monad on standard spaces are precisely the objects of the full subcategory , i.e., standard measurable spaces equipped with a convex structure satisfying a fullness property. The central technique is to view probability measures as -valued functionals and to use the single object as a codense generator, enabling an adjoint factorization of the Giry monad: . The authors construct -algebras on all objects of via the expectation maps and show conversely that every -algebra must lie in , culminating in the equality . This yields a concrete, algebraic, and expectation-centric understanding of probabilistic computation on standard spaces.

Abstract

The Giry monad on the category of measurable spaces restricts to the full subcategory of standard Borel spaces, , which we show is amenable to analysis. contains the space which is the one-point compactification of the real numbers. By viewing probability measures as functionals operating on measurable functions , and taking the restriction of those functionals to operate on affine measurable functions we show that for all object lying in the subcategory of . The objects of are standard spaces with a convex space structure which satisfies the generic ``fullness property''. The morphisms of the category are affine measurable functions. The isomorphism is equivalent to the statement that the full subcategory of consisting of the single object is codense in which allows us to easily construct the -algebras of objects in . This permits an adjoint factorization of the Giry monad as the composite of , which is the Giry monad functor viewed as a functor into , and the partial forgetful functor which forgets the convex space structure. We prove that the category is the category of algebras of the -monad.
Paper Structure (6 sections, 14 theorems, 24 equations, 2 figures)

This paper contains 6 sections, 14 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Restricting the Giry monad to standard spaces
  3. Probability measures as functionals
  4. Probability functionals operating on affine maps are evaluation maps
  5. Constructing $\mathcal{G}$-algebras in $\operatorname{\mathbf{Std}}_{Cvx}$
  6. Showing $\operatorname{\mathbf{Std}}_{Cvx} = \mathbf{Alg}_{\mathcal{G}}$

Key Result

Lemma 1

Let $(X, \sigma(\mathbb{F}))$ be a standard space, and let $P, Q \in \mathcal{G}{X}$. If $P(U) = Q(U)$ for all $U \in \mathbb{F}$ then $P(V) = Q(V)$ for all $V \in \sigma(\mathbb{F})$.

Figures (2)

  • Figure 1: The existence of the $\mathcal{G}$-algebras for objects in $\operatorname{\mathbf{Std}}_{Cvx}$ based upon the codense subcategory $\mathcal{R}$ of $\operatorname{\mathbf{Std}}_{Cvx}$.
  • Figure :

Theorems & Definitions (28)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 18 more