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.

翻译：不精确概率关注于应使用何种概率分布的不确定性问题。它在鲁棒统计学与机器学习中具有应用价值。本文研究用于不精确概率的编程语言模型。我们的核心诉求是希望模型能够支持各类组合——范畴组合与幺半群组合；换言之，以数据流图为指导。另一种等价视角是，我们希望获得一种符合马尔可夫范畴意义的合成概率模型。不精确概率可通过多种方式建模，其中基于单子的主流方法采用概率分布的凸集。该模型不具备完全组合性，因为所涉及的单子不满足交换律，即缺乏恰当的幺半群结构。本工作提出了一种全新的完全组合化建模方案。其核心思想是为非确定性选择赋予名称。为管理名称的重命名与互斥性，我们采用分级单子。我们证明所构建的组合模型具有极大性，并将其与早期单子方法建立关联，论证了该方法能获得更严格的不确定性界限。