We characterize a number of well known systems of approximate inference as loss models: lax sections of 2-fibrations of statistical games, constructed by attaching internally-defined loss functions to Bayesian lenses. Our examples include the relative entropy, which constitutes a strict section, and whose chain rule is formalized by the horizontal composition of the 2-fibration. In order to capture this compositional structure, we first introduce the notion of 'copy-composition', alongside corresponding bicategories through which the composition of copy-discard categories factorizes. These bicategories are a variant of the Copara construction, and so we additionally introduce coparameterized Bayesian lenses, proving that coparameterized Bayesian updates compose optically, as in the non-coparameterized case.

翻译：我们将若干著名的近似推理系统刻画为损失模型：即统计博弈的2-纤维化中的松弛截面，该结构通过将内部定义的损失函数附加到贝叶斯透镜上构建而成。我们的例子包括相对熵（构成一个严格截面），其链式法则由2-纤维化的水平复合形式化。为捕获这种组合结构，我们首先引入"复制-复合"概念及相应的双范畴，通过它们实现复制-丢弃范畴的复合因子化。这些双范畴是Copara构造的变体，因此我们进一步引入共参数化贝叶斯透镜，证明共参数化贝叶斯更新如同非共参数化情形一样，以光学方式组合。