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.

通过 Markov categories 定义无向图模型上的联合概率密度和样本结果的确定性映射的组合的范畴，这是将概率编程和因果推断中的因子化密度的操作定义与概率测度的范畴描述结合的一步。

具分解密度的字符串图

String Diagrams with Factorized Densities

In this paper, we describe a method for estimating the joint probability
density from data samples by assuming that the underlying distribution can be
decomposed as a mixture of product densities with few mixture components. Prior
works have used such a decomposition to estimate the joint density from
lower-dimensional marginals, which can be estimated more reliably with the same
number of samples. We combine two key ideas: dictionaries to represent 1-D
densities, and random projections to estimate the joint distribution from 1-D
marginals, explored separately in prior work. Our algorithm benefits from
improved sample complexity over the previous dictionary-based approach by using
1-D marginals for reconstruction. We evaluate the performance of our method on
estimating synthetic probability densities and compare it with the previous
dictionary-based approach and Gaussian Mixture Models (GMMs). Our algorithm
outperforms these other approaches in all the experimental settings.

本文提出了一种从 1-D 边际概率密度中估计联合概率密度的方法，使用字典表示 1-D 密度和随机投影来估计联合分布，比先前基于字典的方法具有更好的样本复杂度。通过使用 1-D 边缘值进行重构，我们在估计合成概率密度方面优于先前的字典方法和高斯混合模型 (GMMs)。