This paper advances theoretical understanding of infinite-dimensional geometrical properties associated with Bayesian inference. First, we introduce a novel class of infinite-dimensional Hamiltonian systems for saddle Hamiltonian functions whose domains are metric spaces. A flow of this system is generated by a Hamiltonian arc field, an analogue of Hamiltonian vector fields formulated based on (i) the first variation of Hamiltonian functions and (ii) the notion of arc fields that extends vector fields to metric spaces. We establish that this system obeys the conservation of energy. We derive a condition for the existence of the flow, which reduces to local Lipschitz continuity of the first variation under sufficient regularity. Second, we present a system of a Hamiltonian function, called the minimum free energy, whose domain is a metric space of negative log-likelihoods and probability measures. The difference of the posterior and the prior of Bayesian inference is characterised as the first variation of the minimum free energy. Our result shows that a transition from the prior to the posterior defines an arc field on a space of probability measures, which forms a Hamiltonian arc field together with another corresponding arc field on a space of negative log-likelihoods. This reveals the underlying invariance of the free energy behind the arc field.

翻译：本文推进了对贝叶斯推断相关无穷维几何性质的理论理解。首先，我们针对定义域为度量空间的鞍型哈密顿函数，引入一类新型无穷维哈密顿系统。该系统的流由哈密顿弧场生成——这是基于：(i) 哈密顿函数的一阶变分；(ii) 将向量场推广至度量空间的弧场概念，而构建的哈密顿向量场模拟量。我们证明该系统遵循能量守恒定律，并推导出流存在的条件——在充分正则性下该条件可归结为一阶变分的局部利普希茨连续性。其次，我们提出一类称为最小自由能的哈密顿函数系统，其定义域为由负对数似然与概率测度构成的度量空间。贝叶斯推断中后验与先验的差异被表征为最小自由能的一阶变分。我们的结果表明，从先验到后验的转移定义了概率测度空间上的弧场，该弧场与负对数似然空间上对应的另一个弧场共同构成哈密顿弧场。这揭示了弧场背后自由能的隐含不变性。