We introduce the concepts of Bayesian lens, characterizing the bidirectional structure of exact Bayesian inference, and statistical game, formalizing the optimization objectives of approximate inference problems. We prove that Bayesian inversions compose according to the compositional lens pattern, and exemplify statistical games with a number of classic statistical concepts, from maximum likelihood estimation to generalized variational Bayesian methods. This paper is the first in a series laying the foundations for a compositional account of the theory of active inference, and we therefore pay particular attention to statistical games with a free-energy objective.

翻译：我们引入了贝叶斯透镜概念,描述贝叶斯精确推理的双向结构以及统计游戏,正式确定近似推理问题的优化目标。 我们证明贝叶斯反向根据构成透镜模式组成,并以一些经典统计概念作为例子,从最大可能性估算到普遍变异的巴伊斯方法。本文是一系列为积极推理理论的构成说明奠定基础的文件中的第一个,因此我们特别关注以自由能源为目标的统计游戏。