一句话：用命名不确定选择与分级单子，在 Markov 范畴中为不精确概率给出全组合语义，并获得更紧界。
不精确概率关注应选用哪个概率分布的不确定性，广泛用于稳健统计与机器学习。本文研究不精确概率的编程语言模型，目标是支持范畴式与幺半式等各种组合，并符合 Markov 范畴意义下的合成概率。已有基于凸概率分布集的单子模型因单子不可交换、缺乏真正幺半结构，组合性不足。作者提出为不确定选择命名，并用分级单子（graded monads）管理重命名与名称不交性，从而得到全组合模型。作者证明该模型是极大的，并建立其与早期单子方法的关系，说明能得到更紧的不确定性界。
Imprecise probability is concerned with uncertainty about which probability distributions to use. It has applications in robust statistics and machine learning. We look at programming language models for imprecise probability. Our desiderata are that we would like our model to support all kinds of composition, categorical and monoidal; in other words, guided by dataflow diagrams. Another equivalent perspective is that we would like a model of synthetic probability in the sense of Markov categories. Imprecise probability can be modelled in various ways, with the leading monad-based approach using convex sets of probability distributions. This model is not fully compositional because the monad involved is not commutative, meaning it does not have a proper monoidal structure. In this work, we provide a new fully compositional account. The key idea is to name the non-deterministic choices. To manage the renamings and disjointness of names, we use graded monads. We show that the resulting compositional model is maximal and relate it with the earlier monadic approach, proving that we obtain tighter bounds on the uncertainty.
@article{paperbot3244,
title = {Compositional Imprecise Probability: A Solution from Graded Monads and Markov Categories},
author = {Jack Liell-Cock and Sam Staton},
journal = {Proceedings of the ACM on Programming Languages},
volume = {9},
number = {POPL},
year = {2025},
doi = {10.1145/3704890}
}