This dissertation reports some first steps towards a compositional account of active inference and the Bayesian brain. Specifically, we use the tools of contemporary applied category theory to supply functorial semantics for approximate inference. To do so, we define on the `syntactic' side the new notion of Bayesian lens and show that Bayesian updating composes according to the compositional lens pattern. Using Bayesian lenses, and inspired by compositional game theory, we define fibrations of statistical games and classify various problems of statistical inference as corresponding sections: the chain rule of the relative entropy is formalized as a strict section, while maximum likelihood estimation and the free energy give lax sections. In the process, we introduce a new notion of `copy-composition'. On the `semantic' side, we present a new formalization of general open dynamical systems (particularly: deterministic, stochastic, and random; and discrete- and continuous-time) as certain coalgebras of polynomial functors, which we show collect into monoidal opindexed categories (or, alternatively, into algebras for multicategories of generalized polynomial functors). We use these opindexed categories to define monoidal bicategories of cilia: dynamical systems which control lenses, and which supply the target for our functorial semantics. Accordingly, we construct functors which explain the bidirectional compositional structure of predictive coding neural circuits under the free energy principle, thereby giving a formal mathematical underpinning to the bidirectionality observed in the cortex. Along the way, we explain how to compose rate-coded neural circuits using an algebra for a multicategory of linear circuit diagrams, showing subsequently that this is subsumed by lenses and polynomial functors.

翻译：本论文报告了关于主动推理与贝叶斯大脑组合解释的初步研究成果。具体而言，我们运用当代应用范畴论工具为近似推理提供函子语义。为此，我们在"语法"层面定义了贝叶斯透镜这一新概念，并证明贝叶斯更新遵循组合透镜模式进行组合。受到组合博弈论的启发，我们利用贝叶斯透镜构建了统计博弈的纤维化，并将各类统计推理问题分类为相应的截面：相对熵的链式法则被形式化为严格截面，而极大似然估计与自由能则给出松弛截面。在此过程中，我们引入"复制-组合"这一新概念。在"语义"层面，我们提出了一种开放动力系统（包括：确定性、随机性与随机性系统；离散时间与连续时间系统）的新形式化方法，将其表示为多项式函子的余代数，并证明这些余代数可构成幺半群opindexed范畴（或等价地，可构成广义多项式函子多范畴上的代数）。我们利用这些opindexed范畴定义纤毛的幺半群双范畴：即控制透镜的动力系统，并为函子语义提供靶标。据此，我们构建了相应函子，以解释自由能原理下预测编码神经回路中的双向组合结构，从而为大脑皮层中观察到的双向性提供严谨的数学基础。在此过程中，我们还阐释如何利用线性电路图多范畴上的代数来组合率编码神经回路，并随后证明这一方法可被透镜与多项式函子所涵盖。