Prediction is a central problem in Statistics, and there is currently a renewed interest for the so-called predictive approach in Bayesian statistics. What is the latter about? One has to return on foundational concepts, which we do in this paper, moving from the role of exchangeability and reviewing forms of partial exchangeability for more structured data, with the aim of discussing their use and implications in Bayesian statistics. There we show the underlying concept that, in Bayesian statistics, a predictive rule is meant as a learning rule - how one conveys past information to information on future events. This concept has implications on the use of exchangeability and generally invests all statistical problems, also in inference. It applies to classic contexts and to less explored situations, such as the use of predictive algorithms that can be read as Bayesian learning rules. The paper offers a historical overview, but also includes a few new results, presents some recent developments and poses some open questions.

翻译：预测是统计学中的核心问题，当前贝叶斯统计学中对所谓的预测方法再次产生浓厚兴趣。后者究竟指什么？我们需要回归基础概念，这正是本文所做的工作——从可交换性的作用出发，评述针对更结构化数据的部分可交换性形式，旨在探讨其在贝叶斯统计学中的使用与含义。我们在此揭示了一个基本概念：在贝叶斯统计学中，预测规则本质上是一种学习规则——如何将过去信息传递到未来事件信息。这一概念对可交换性的使用具有影响，并普遍涉及所有统计问题（包括推断问题）。它既适用于经典场景，也适用于较少探索的情况，例如可被视为贝叶斯学习规则的预测算法应用。本文不仅提供了历史概述，还包含若干新结果、介绍了一些最新进展，并提出了一些开放性问题。