发表机构
The Ohio State University; Amazon.com Inc.（俄亥俄州立大学; 亚马逊公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究概率单纯形上的小步主动推理，提出基于相对自由能和动能约束的序列变分原理，证明小步贪心更新全局最优，并给出指数倾斜闭式解。
AI 中文摘要
本文是两部分关于信息几何物理研究中的第二篇。第一部分为概率单纯形上的分布运动建立了物理基础，而本文研究该框架如何在主动推理中体现。本文的论述完全自包含，不需要熟悉第一部分。我们特别关注通过小分布步长进行的主动推理以及这种局部运动所诱导的几何结构。从初始分布出发，智能体通过一系列受约束的更新，将其信念状态演化到最终目标分布。我们定义了相对于偏好分布的相对自由能泛函，并将其扩展为类似于亥姆霍兹/吉布斯自由能分解的相对势能。演化受到每一步动能约束的限制，该约束通过连续分布之间的库尔贝克-莱布勒（KL）散度表达，作为概率单纯形上的离散动能。利用KL球上的信息几何勾股定理，我们证明了足够小的局部移动优于大的直接跳跃，并且在动能约束下，贪心最大化自由能减少是全局最优的。这导致了一个序列变分原理，其中最优轨迹最小化优化问题的相应拉格朗日量。与经典力学类似，拉格朗日量采取动能与势能之差的形式，为单纯形上的分布运动建立了最小作用量原理。所得的最优更新具有闭式解，即当前分布向偏好分布的指数倾斜版本，由逆温度类乘子参数化。我们进一步将框架扩展以纳入状态依赖的测地线...
英文摘要
This paper is the second in a two-part investigation of the physics of information geometry. While Part I develops a physical foundation for distributional motion on the probability simplex, the present paper studies how that framework manifests in active inference. The treatment is fully self-contained and does not require familiarity with Part I. We focus in particular on active inference through small distributional steps and the geometric structure induced by such local motion. Starting from an initial distribution, an agent evolves its belief state toward a final target distribution through a sequence of constrained updates. We define a relative free energy functional with respect to the preferred distribution and extend it to a relative potential energy analogous to the Helmholtz/Gibbs free-energy decomposition. The evolution is subject to a per-step kinetic constraint expressed through the Kullback-Leibler (KL) divergence between consecutive distributions, which serves as a discrete kinetic energy on the probability simplex. Using the information-geometric Pythagorean theorem on KL balls, we show that sufficiently small local moves dominate large direct jumps, and that greedy maximization of free-energy reduction is globally optimal under the kinetic constraint. This leads to a sequential variational principle in which the optimal trajectory minimizes the associated Lagrangian of the optimization problem. Similar to classical mechanics, the Lagrangian takes on the form as the difference between the kinetic and potential terms, establishing a least-action principle for distributional motion on the simplex. The resulting optimal update admits a closed form as an exponentially tilted version of the current distribution toward the preferred distribution, parametrized by an inverse-temperature-like multiplier. We further extend the framework to incorporate state-dependent geodesic...
