Disintegration and Bayesian inversion via string diagrams

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.

Categorical Stochastic Processes and Likelihood

“…One of the largest differences between this construction and those of Cho and Jacobs [4] and Culbertson and Sturtz [6] is the treatment of model updates in the face of new data. While these authors also describe categorical frameworks in which we can model how a new observation updates the parameters of a statistical model, they primarily study Bayesian algorithms in which the model parameters are represented with a probability distribution.…”

“…Separately, one of the most active areas of applied category theory focuses on building a categorical framework for probability theory and statistics. Researchers like Fritz [14], Cho and Jacobs [4], and Culbertson and Sturtz [6; 7] have developed strategies for describing the construction of probabilistic models from data in categorical terms. We aim to bridge these streams of research by using a probabilistic construction to define an optimization objective.…”

“…Cho and Jacobs [4] and Culbertson and Sturtz [6;7] explore how new data points affect their models' epistemic uncertainty, or uncertainty due to limited data or knowledge. For example, a simple model of a complex nonlinear system is likely to have high epistemic uncertainty.…”

Infinite products and zero-one laws in categorical probability

“…Returning to the general theory, in terms of string diagrams the unique morphism del X : X → I for an object X ∈ C is denoted by X The projection maps id ⊗ del Y : X ⊗ Y → X and del X ⊗ id : X ⊗ Y → Y , which are the ones from (1), are correspondingly written as Y X Y X Our goal is to use semicartesian monoidal categories in order to develop aspects of probability theory in categorical terms, in such a way that instantiating this theory in Stoch or BorelStoch recovers the standard theory. As it turns out, doing so requires a bit more structure, in a form which has been axiomatized first by Cho and Jacobs [3] as affine CD-categories, although very similar definitions occur in earlier work of Golubtsov [10]. We here follow the more intuitive terminology of our own [6, Definition 2.1].…”

“…Markov categories are an approach to the foundations of probability and statistics based on category theory, proposed first by Golubtsov [10], rediscovered recently independently by Cho and Jacobs [3], and developed extensively by the first-named author in [6]. The basic observation is that Markov kernels can be composed sequentially and in parallel, making them into a symmetric monoidal category Stoch.…”