Markov categories, having tensors with copying and discarding, provide a setting for categorical probability. This paper uses finite colimits and what we call uniform states in such Markov categories to define a (fixed size) multiset functor, with basic operations for sums and zips of multisets, and a graded monad structure. Multisets can be used to represent both urns filled with coloured balls and also draws of multiple balls from such urns. The main contribution of this paper is the abstract definition of multinomial and hypergeometric distributions on multisets, as draws. It is shown that these operations interact appropriately with various operations on multisets.
翻译:Markov 类别, 带有复制和丢弃的抗拉器, 提供了绝对概率的设定。 本文使用限值共同限值和我们称之为统一的 Markov 类国家来定义一个( 固定大小) 多立方的真菌( 多立方体), 其总和和拉链的基本操作, 以及一个分级的monad 结构 。 多立方位可以用来代表填充有色球的骨质, 以及从这些骨质中提取多个球 。 本文的主要贡献是多立方位( 绘图) 的多音量和超几何分布的抽象定义 。 显示这些操作与多立方形上的各种操作适当互动 。