On Recursive Bayesian Predictive Distributions

A Bayesian framework is attractive in the context of prediction, but a fast recursive update of the predictive distribution has apparently been out of reach, in part because Monte Carlo methods are generally used to compute the predictive. This article shows that online Bayesian prediction is possible by characterizing the Bayesian predictive update in terms of a bivariate copula, making it unnecessary to pass through the posterior to update the predictive. In standard models, the Bayesian predictive update corresponds to familiar choices of copula but, in nonparametric problems, the appropriate copula may not have a closed-form expression. In such cases, our new perspective suggests a fast recursive approximation to the predictive density, in the spirit of Newton’s predictive recursion algorithm, but without requiring evaluation of normalizing constants. Consistency of the new algorithm is shown, and numerical examples demonstrate its quality performance in finite-samples compared to fully Bayesian and kernel methods. Supplementary materials for this article are available online.贝叶斯框架在预测任务中颇具吸引力，但长期以来，快速递归更新预测分布似乎难以实现，部分原因在于预测分布的计算通常依赖蒙特卡洛（Monte Carlo）方法。本文通过以二元连接函数（bivariate copula）刻画贝叶斯预测更新过程，证明了在线贝叶斯预测的可行性，无需借助后验分布即可完成预测更新。在标准模型中，贝叶斯预测更新对应于常见的连接函数选择，但在非参数问题中，合适的连接函数可能不存在闭式表达式。针对此类场景，本文提出的全新视角借鉴了牛顿预测递归算法（Newton’s predictive recursion algorithm）的思想，可实现预测密度的快速递归近似，且无需计算归一化常数。本文证明了该新算法的一致性，并通过数值示例表明，相较于全贝叶斯方法与核方法，其在有限样本下的表现优异。本文的补充材料可在线获取。