Bayesian methods are often optimal, yet increasing pressure for fast computations, especially with streaming data, brings renewed interest in faster, possibly suboptimal, solutions. The extent to which these algorithms approximate Bayesian solutions is a question of interest, but often unanswered. We propose a methodology to address this question in predictive settings, when the algorithm can be reinterpreted as a probabilistic predictive rule. We specifically develop the proposed methodology for a recursive procedure for on‐line learning in non‐parametric mixture models, which is often referred to as Newton's algorithm. This algorithm is simple and fast; however, its approximation properties are unclear. By reinterpreting it as a predictive rule, we can show that it underlies a statistical model which is, asymptotically, a Bayesian, exchangeable mixture model. In this sense, the recursive rule provides a quasi‐Bayes solution. Although the algorithm offers only a point estimate, our clean statistical formulation enables us to provide the asymptotic posterior distribution and asymptotic credible intervals for the mixing distribution. Moreover, it gives insights for tuning the parameters, as we illustrate in simulation studies, and paves the way to extensions in various directions. Beyond mixture models, our approach can be applied to other predictive algorithms.

中文翻译：

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混合模型中顺序学习过程的拟贝叶斯性质

贝叶斯方法通常是最佳的，但是对于快速计算（尤其是流数据）的压力越来越大，这引起了人们对更快，可能不是最佳解决方案的兴趣。这些算法在多大程度上近似贝叶斯解决方案是一个令人感兴趣的问题，但常常没有答案。当算法可以重新解释为概率预测规则时，我们提出了一种在预测设置中解决此问题的方法。我们专门为非参数混合模型中的在线学习的递归过程开发了建议的方法，该方法通常被称为牛顿算法。该算法简单快捷。但是，其近似性质尚不清楚。通过将其重新解释为预测规则，我们可以证明它是统计模型的基础，该统计模型渐近地为贝叶斯模型，可交换混合物模型。从这个意义上讲，递归规则提供了一种拟贝叶斯解决方案。尽管该算法仅提供一个点估计，但我们干净的统计公式使我们能够为混合分布提供渐近后验分布和渐近可信区间。此外，正如我们在仿真研究中所说明的那样，它为调整参数提供了见识，并为在各个方向上的扩展铺平了道路。除了混合模型之外，我们的方法还可以应用于其他预测算法。正如我们在仿真研究中所说明的那样，它为调整参数提供了见识，并为向各个方向扩展铺平了道路。除了混合模型之外，我们的方法还可以应用于其他预测算法。正如我们在模拟研究中所说明的那样，它为调整参数提供了见识，并为在各个方向上的扩展铺平了道路。除了混合模型之外，我们的方法还可以应用于其他预测算法。