Given a sequence $X=(X_1,X_2,\ldots)$ of random observations, a Bayesian forecaster aims to predict $X_{n+1}$ based on $(X_1,\ldots,X_n)$ for each $n\ge 0$. To this end, in principle, she only needs to select a collection $\sigma=(\sigma_0,\sigma_1,\ldots)$, called ``strategy" in what follows, where $\sigma_0(\cdot)=P(X_1\in\cdot)$ is the marginal distribution of $X_1$ and $\sigma_n(\cdot)=P(X_{n+1}\in\cdot\mid X_1,\ldots,X_n)$ the $n$-th predictive distribution. Because of the Ionescu-Tulcea theorem, $\sigma$ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability is to be selected. In a nutshell, this is the predictive approach to Bayesian learning. A concise review of the latter is provided in this paper. We try to put such an approach in the right framework, to make clear a few misunderstandings, and to provide a unifying view. Some recent results are discussed as well. In addition, some new strategies are introduced and the corresponding distribution of the data sequence $X$ is determined. The strategies concern generalized P\'olya urns, random change points, covariates and stationary sequences.

翻译：给定随机观测序列 $X=(X_1,X_2,\ldots)$，贝叶斯预测者旨在基于 $(X_1,\ldots,X_n)$ 对每个 $n\ge 0$ 预测 $X_{n+1}$。为此，原则上她只需选取一个集合 $\sigma=(\sigma_0,\sigma_1,\ldots)$（下文称为"策略"），其中 $\sigma_0(\cdot)=P(X_1\in\cdot)$ 是 $X_1$ 的边缘分布，而 $\sigma_n(\cdot)=P(X_{n+1}\in\cdot\mid X_1,\ldots,X_n)$ 是第 $n$ 步预测分布。根据 Ionescu-Tulcea 定理，$\sigma$ 可直接赋值，无需经过通常的先验/后验框架。其主要优势在于无需选择先验概率。简而言之，这就是贝叶斯学习的预测性方法。本文对该方法进行了简洁综述。我们试图将此类方法置于正确框架中，澄清若干误解，并提供统一的视角。文中还讨论了一些最新成果。此外，本文引入若干新策略，并确定了相应数据序列 $X$ 的分布。这些策略涉及广义 Pólya 瓮模型、随机变点、协变量及平稳序列。