# Categorical Stochastic Processes and Likelihood

arXiv: https://arxiv.org/abs/2005.04735 · HTML: https://mlsanity.com/paper/2005.04735/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 May 2020.

## Built on by (most cited first, 6 of 6)

- [Categorical Foundations of Gradient-Based Learning](https://mlsanity.com/paper/2103.01931/) (2021)
- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [Deep Learning with Parametric Lenses](https://mlsanity.com/paper/2404.00408/) (2024)
- [A Categorical Analysis of Large Language Models and Why LLMs Circumvent the Symbol Grounding Problem](https://mlsanity.com/paper/2512.09117/) (2025)
- [Compositional Active Inference II: Polynomial Dynamics. Approximate Inference Doctrines](https://mlsanity.com/paper/2208.12173/) (2022)
- [To Describe or Construct Statistical Learning Models Using the Category-theoretical Language](https://mlsanity.com/paper/2608.03706/) (2026)

## Abstract (arXiv)

In this work we take a Category Theoretic perspective on the relationship between probabilistic modeling and function approximation. We begin by defining two extensions of function composition to stochastic process subordination: one based on the co-Kleisli category under the comonad (Omega x -) and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category Stoch and other Markov Categories. Next, we apply the Para construction to extend stochastic processes to parameterized statistical models and we define a way to compose the likelihood functions of these models. We conclude with a demonstration of how the Maximum Likelihood Estimation procedure defines an identity-on-objects functor from the category of statistical models to the category of Learners. Code to accompany this paper can be found at https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood