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A categorical foundation for Bayesian probability (1205.1488v3)

Published 7 May 2012 in math.CT and math.PR

Abstract: Given two measurable spaces $H$ and $D$ with countably generated $\sigma$-algebras, a perfect prior probability measure $P_H$ on $H$ and a sampling distribution $S: H \rightarrow D$, there is a corresponding inference map $I: D \rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $\mu: 1 \rightarrow D$, a posterior probability $\widehat{P_H}= I \circ \mu$ can be computed. This procedure is iterative: with each updated probability $P_H$, we obtain a new joint distribution which in turn yields a new inference map $I$ and the process repeats with each additional measurement. The main result uses an existence theorem for regular conditional probabilities by Faden, which holds in more generality than the setting of Polish spaces. This less stringent setting then allows for non-trivial decision rules (Eilenberg--Moore algebras) on finite (as well as non finite) spaces, and also provides for a common framework for decision theory and Bayesian probability.

Citations (49)

Summary

  • The paper establishes a categorical foundation for Bayesian probability using perfect probabilistic mappings, building on prior work in category theory.
  • It introduces the category of perfect probabilistic mappings (**P**) on countably generated measurable spaces, which facilitates the derivation of joint distributions and regular conditional probabilities essential for Bayesian inference.
  • This framework offers a robust method for integrating decision-making processes with Bayesian models, providing theoretical advancements and potential for novel applications in complex systems.

A Categorical Foundation for Bayesian Probability: An Overview

In "A Categorical Foundation for Bayesian Probability," Culbertson and Sturtz offer a comprehensive exploration of Bayesian probability through the lens of category theory. This paper notably builds on earlier work by Lawvere, Giry, and others to establish a cohesive categorical framework suitable for Bayesian inference, thus advancing the theoretical underpinnings of probabilistic mapping and decision theory without the restrictive assumptions often relied upon in Polish spaces.

Background and Motivation

Category theory has served as a fertile ground for formalizing various constructs in mathematics, including probabilistic mappings. The paper acknowledges Lawvere's pioneering efforts in defining the Kleisli category of probabilistic mappings and Giry's further expansion of these concepts. It seeks to address the gap in leveraging categorical constructs for Bayesian probability, an area where previous works focused predominantly on non-Bayesian inference.

Foundations and Main Contributions

Central to this paper is the innovative approach to Bayesian probability using the category of perfect probabilistic mappings (denoted as P). This category operates over countably generated measurable spaces equipped with perfect probability measures, ensuring the existence of regular conditional probabilities, which are foundational for Bayesian inference.

  1. Perfect Probabilistic Mappings: The authors define the category P with objects being countably generated measurable spaces and morphisms as parametrized families of perfect probability measures. This category circumvents issues faced in topological approaches by not restricting itself to Polish spaces.
  2. Joint and Marginal Distributions: The paper meticulously elaborates on the derivation of joint distributions using conditional probabilities, demonstrating how the category P facilitates such constructions. By utilizing morphisms called conditionals, a unique joint distribution can be determined under compatibility conditions, successfully capturing the essence of Bayesian inference.
  3. Existence of Regular Conditional Probabilities: Culbertson and Sturtz provide an existence theorem for regular conditional probabilities within their categorical framework, pivotal for inference processes in Bayesian probability. This theorem supports the commutative structure of inference in the category P.
  4. Bayesian Probability via Category Theory: By applying categorical methods, the authors explicate Bayesian processes as iterative updates of probability models, leveraging prior probabilities, sampling distributions, and posterior probabilities across different inference steps.

Implications and Future Directions

The categorical framework presented by Culbertson and Sturtz has practical and theoretical implications:

  • Practical Utility: This approach streamlines the integration of decision-making processes with Bayesian inference models, offering clarity in scenarios where data collection and hypothesis testing operate under conditions of uncertainty. The categorical perspective enhances modeling strategies in complex systems requiring iterative inference and adjustment.
  • Theoretical Advancements: This paper provides a crucial stepping stone towards unifying Bayesian probability models and decision-theoretic processes through a categorical lens. By encompassing regular conditional probabilities, it sets the stage for new research directions in developing robust probabilistic models that eschew unnecessary topological constraints.

The authors have opened avenues for further exploration in areas such as higher-order distributions, the honing of decision rules without continuity constraints, and enhanced interplays between various categorical structures. Future studies might explore characterizing decision rules within this categorical framework or expand the applications of Bayesian probability via other categorical constructs.

In sum, "A Categorical Foundation for Bayesian Probability" represents a significant contribution to the domain of probabilistic category theory, offering a well-defined infrastructure for Bayesian inference that promises innovations both in practical applications and foundational understanding. As such, it is a pivotal reference point for researchers seeking to extend Bayesian methodologies within categorical frameworks.

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