Disintegration and Bayesian Inversion via String Diagrams (1709.00322v3)

Abstract: The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.

• The paper demonstrates how disintegration and Bayesian inversion are formulated using string diagrams to model conditional probability abstractly.
• It establishes conditions for disintegration existence in discrete and measure-theoretic settings and introduces 'almost equality' to resolve uniqueness issues.
• Practical examples, including naive Bayesian classification via the EfProb tool, illustrate the utility of this categorical approach in machine learning and statistical modeling.

Overview of Disintegration and Bayesian Inversion via String Diagrams

The paper "Disintegration and Bayesian Inversion via String Diagrams" by Kenta Cho and Bart Jacobs explores the concepts of disintegration and Bayesian inversion within the framework of string diagrams in category theory. These notions form fundamental components of conditional probability theory and have notable applications in both discrete and measure-theoretic probability realms. This work situates these concepts within an abstract graphical setting, yielding coherent formulations that facilitate the derivation of foundational results in conditional probability.

Disintegration refers to the derivation of a conditional probability (or a channel) from a joint probability state, while Bayesian inversion involves reversing a channel to infer prior probabilities from posterior states. Here, the authors leverage string diagrams, a graphical language for symmetric monoidal categories, to abstractly describe probability and statistics, thereby eschewing the need for a specific probability space.

Key Results

The paper presents several strong results:

1. Existence of Disintegration: It demonstrates the conditions under which disintegration can be formulated in both discrete and continuous settings, emphasizing that while disintegrations are readily obtainable in the discrete context via standard definitions of conditional probability, they are notably complex in measure-theoretic settings where their existence is not necessarily guaranteed.
2. Bayesian Inversion: A substantial exploration of how Bayesian inversion, an operation essential for backward inference, is effectively modeled using string diagrams. It is shown that disintegration and Bayesian inversion are closely linked and can derive one from the other under specified conditions.
3. Almost Equality: The authors introduce and utilize a notion of 'almost equality' to solve issues of disintegration uniqueness that arise in nondeterministic measure-theoretic contexts.
4. Graphical Language for Independence: The work significantly extends to using these abstract tools to describe conditional independencies, reflecting these concepts through graphical axioms and transformations akin to graphoid properties, essential for reasoning about dependencies in Bayesian networks.
5. Practical Applications: Leveraging the EfProb tool, examples such as naive Bayesian classification highlight the utility of disintegration and inversion in concrete machine learning tasks.

Implications and Future Directions

The implications of this paper are multifaceted. The graphical treatment via string diagrams illustrates a powerful way to conceptualize and manipulate probabilities that transcends traditional symbolic calculations. It offers a categorical framework capable of capturing richer probabilistic structures, including those pertinent to Bayesian learning and quantum foundations.

Theoretically, the paper provides foundational insights that could influence the development of probabilistic programming languages and abstractions. Practically, it suggests potential improvements in the design and understanding of statistical models that rely upon complex dependency structures and evidential reasoning, such as those encountered in causal inference and probabilistic graphical models.

Future developments may include extending this framework to incorporate more general forms of nondeterministic and possibly non-finite measures or establishing more robust connections to other categorical structures encountering in contemporary AI problems, such as quantum computing or distributed systems.

Overall, this paper sets a robust precedent for integrating categorical methods and probabilistic reasoning, opening doors for more axiomatic treatments of complex probabilistic models.