Categorical distribution theory; heat equation

图极限的范畴方法

Categorical approach to graph limits

本文提出图极限的范畴框架，通过定义对象和态射，结合范畴论工具证明图极限空间的紧性，并引入基于s-收敛的收敛概念。

Towards probabilistic partial metric spaces: Diagonals between distance distributions

The quantale of distance distributions is of fundamental importance for understanding probabilistic metric spaces as enriched categories. Motivated by the categorical interpretation of partial metric spaces, we are led to investigate the quantaloid of diagonals between distance distributions, which is expected to establish the categorical foundation of probabilistic partial metric spaces. Observing that the quantale of distance distributions w.r.t. an arbitrary continuous t-norm is non-divisible, we precisely characterize diagonals between distance distributions, and prove that one-step functi…

Conditional Distributions for Quantum Systems

Conditional distributions, as defined by the Markov category framework, are studied in the setting of matrix algebras (quantum systems). Their construction as linear unital maps are obtained via a categorical Bayesian inversion procedure. Simple criteria establishing when such linear maps are positive are obtained. Several examples are provided, including the standard EPR scenario, where the EPR correlations are reproduced in a purely compositional (categorical) manner. A comparison between the Bayes map, the Petz recovery map, and the Leifer-Spekkens acausal belief propagation is provided, il…

用概率范畴框架对基因调控网络建模

Modeling GRNs with a Probabilistic Categorical Framework

本文提出PC-GRN框架，结合范畴论、贝叶斯类型化Petri网、GFlowNet和HyperNetwork，直接从数据学习基因调控网络结构与参数的后验分布，实现严格且可解释的不确定性建模。

Aldous-Hoover定理在范畴概率中的应用

The Aldous$\unicode{x2013}$Hoover Theorem in Categorical Probability

本文从Markov范畴理论出发，证明了Aldous-Hoover定理，并引入了Cauchy-Schwarz公理，同时探讨了其在分层可交换性中的应用。

A categorical foundation for Bayesian probability

Given two measurable spaces $H$ and $D$ with countably generated $σ$-algebras, a perfect prior probability measure $P_H$ on $H$ and a sampling distribution $S: H \rightarrow D$, there is a corresponding inference map $I: D \rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $μ: 1 \rightarrow D$, a posterior probability $\widehat{P_H}= I \circ μ$ can be computed. This procedure is iterative: with each updated probability $P_H$, we obtain a new joint distribution which in turn yields a new inference map $I$ and the process repeats with each additional measu…