Bayesian nonparametric mixture models provide a flexible framework for data analysis but are often hindered by the computational expense of traditional inference methods like MCMC. A fast, recursive algorithm proposed by Newton (2002) offers a practical alternative, yet its formal connection to Bayesian inference and its theoretical properties remain only partially understood. This paper reveals a new geometric interpretation of this class of predictive recursions. We demonstrate that Newton's recursion is a discrete-time approximation of a gradient flow on the space of probability measures governed by the Fisher-Rao geometry, providing the first rigorous dynamical characterisation of this family of estimators. This geometric perspective provides a principled theoretical foundation for studying these recursions: it clarifies their convergence behaviour, situates them within the variational Bayes literature, and yields a systematic basis for generalisation by modifying the underlying geometry and discretisation. In contrast to approaches that construct gradient flows from a prescribed variational objective, this work proceeds in the reverse direction: beginning from an existing recursive estimator and uncovering the variational problem it implicitly solves, it opens a pathway for the systematic analysis and extension of a broad class of sequential Bayesian estimators.

翻译：贝叶斯非参数混合模型为数据分析提供了灵活框架，但传统推断方法（如MCMC）的计算代价常制约其应用。Newton（2002）提出的快速递归算法提供了实用的替代方案，然而其与贝叶斯推断的形式化联系及理论性质仍仅被部分理解。本文揭示了这类预测递归的新几何解释：我们证明Newton递归是Fisher-Rao几何主导的概率测度空间上梯度流的离散时间近似，首次为该类估计量建立了严谨的动力学刻画。这一几何视角为研究这些递归提供了规范的理论基础：阐明了其收敛行为，将其置于变分贝叶斯文献的语境中，并通过修正底层几何结构与离散化方法为泛化提供了系统性依据。不同于从预设变分目标构建梯度流的传统方法，本研究反向推进：从现有递归估计器出发挖掘其隐含求解的变分问题，为系统性分析与扩展广泛类别的序贯贝叶斯估计器开辟了路径。