This paper makes two theoretical contributions. First, we establish a novel class of Hamiltonian systems, called arc Hamiltonian systems, for saddle Hamiltonian functions over infinite-dimensional metric spaces. Arc Hamiltonian systems generate a flow that satisfies the law of conservation of energy everywhere in a metric space. They are governed by an extension of Hamilton's equation formulated based on (i) the framework of arc fields and (ii) an infinite-dimensional gradient, termed the arc gradient, of a Hamiltonian function. We derive conditions for the existence of a flow generated by an arc Hamiltonian system, showing that they reduce to local Lipschitz continuity of the arc gradient under sufficient regularity. Second, we present two Hamiltonian functions, called the cumulant generating functional and the centred cumulant generating functional, over a metric space of log-likelihoods and measures. The former characterises the posterior of Bayesian inference as a part of the arc gradient that induces a flow of log-likelihoods and non-negative measures. The latter characterises the difference of the posterior and the prior as a part of the arc gradient that induces a flow of log-likelihoods and probability measures. Our results reveal an implication of the belief updating mechanism from the prior to the posterior as an infinitesimal change of a measure in the infinite-dimensional Hamiltonian flows.

翻译：本文提出两项理论贡献。首先，针对无穷维度量空间上的鞍点哈密顿函数，我们建立了一类新型哈密顿系统——弧哈密顿系统。该系统的生成流在度量空间处处满足能量守恒定律，其动力学由基于(i)弧场框架与(ii)哈密顿函数的无穷维梯度（称为弧梯度）扩展的哈密顿方程调控。我们推导了弧哈密顿系统生成流的存在条件，证明在充分正则性条件下该条件可归约为弧梯度的局部Lipschitz连续性。其次，在似然函数与测度构成的度量空间上，我们定义了两类哈密顿函数——累积量生成泛函与中心累积量生成泛函。前者通过表征似然函数与非负测度流的弧梯度分量，刻画了贝叶斯推断的后验分布；后者通过表征似然函数与概率测度流的弧梯度分量，揭示了后验分布与先验分布的差异。研究结果表明，从先验到后验的信念更新机制本质上可解释为无穷维哈密顿流中测度的无穷小变化。