# String Diagrams with Factorized Densities

arXiv: https://arxiv.org/abs/2305.02506 · HTML: https://mlsanity.com/paper/2305.02506/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 44 references that are in the map.
- First posted on arXiv 4 May 2023.

## Builds on (most cited first, 19 of 19)

- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review](https://mlsanity.com/paper/1805.00909/) (2018)
- [Elements of Sequential Monte Carlo](https://mlsanity.com/paper/1903.04797/) (2019)
- [An Introduction to Probabilistic Programming](https://mlsanity.com/paper/1809.10756/) (2018)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [Nested Sequential Monte Carlo Methods](https://mlsanity.com/paper/1502.02536/) (2015)
- [A Convenient Category for Higher-Order Probability Theory](https://mlsanity.com/paper/1701.02547/) (2017)
- [The d-separation criterion in Categorical Probability](https://mlsanity.com/paper/2207.05740/) (2022)
- [Denotational validation of higher-order Bayesian inference](https://mlsanity.com/paper/1711.03219/) (2017)
- [Free gs-monoidal categories and free Markov categories](https://mlsanity.com/paper/2204.02284/) (2022)
- [Affine Monads and Lazy Structures for Bayesian Programming](https://mlsanity.com/paper/2212.07250/) (2022)
- [Rewriting modulo symmetric monoidal structure](https://mlsanity.com/paper/1602.06771/) (2016)
- [Discrete-Continuous Mixtures in Probabilistic Programming: Generalized Semantics and Inference Algorithms](https://mlsanity.com/paper/1806.02027/) (2018)
- [Notes on Category Theory with examples from basic mathematics](https://mlsanity.com/paper/1912.10642/) (2019)
- [The Base Measure Problem and its Solution](https://mlsanity.com/paper/2010.09647/) (2020)
- [Probabilistic Programming Semantics for Name Generation](https://mlsanity.com/paper/2007.08638/) (2020)
- [Compositionality in algorithms for smoothing](https://mlsanity.com/paper/2303.13865/) (2023)
- [Data-Parallel Algorithms for String Diagrams](https://mlsanity.com/paper/2305.01041/) (2023)

## Abstract (arXiv)

A growing body of research on probabilistic programs and causal models has highlighted the need to reason compositionally about model classes that extend directed graphical models. Both probabilistic programs and causal models define a joint probability density over a set of random variables, and exhibit sparse structure that can be used to reason about causation and conditional independence. This work builds on recent work on Markov categories of probabilistic mappings to define a category whose morphisms combine a joint density, factorized over each sample space, with a deterministic mapping from samples to return values. This is a step towards closing the gap between recent category-theoretic descriptions of probability measures, and the operational definitions of factorized densities that are commonly employed in probabilistic programming and causal inference.