In probabilistic updating one transforms a prior distribution in the light of given evidence into a posterior distribution, via what is called conditioning, updating, belief revision or inference. This is the essence of learning, as Bayesian updating. It will be illustrated via a physical model involving (adapted) water flows through pipes with different diameters. Bayesian updating makes us wiser, in the sense that the posterior distribution makes the evidence more likely than the prior, since it incorporates the evidence. Things are less clear when one wishes to learn from multiple pieces of evidence / data. It turns out that there are (at least) two forms of updating for this, associated with Jeffrey and Pearl. The difference is not always clearly recognised. This paper provides an introduction and an overview in the setting of discrete probability theory. It starts from an elementary question, involving multiple pieces of evidence, that has been sent to a small group academic specialists. Their answers show considerable differences. This is used as motivation and starting point to introduce the two forms of updating, of Jeffrey and Pearl, for multiple inputs and to elaborate their properties. In the end the account is related to so-called variational free energy (VFE) update in the cognitive theory of predictive processing. It is shown that both Jeffrey and Pearl outperform VFE updating and that VFE updating need not decrease divergence - that is correct errors - as it is supposed to do.

翻译：在概率更新中，人们通过称为条件化、更新、信念修正或推理的方式，根据给定证据将先验分布转化为后验分布。这是学习的本质，即贝叶斯更新。本文将借助一个涉及不同直径管道内（经适配的）水流的物理模型加以阐释。贝叶斯更新使我们更明智，因为后验分布使证据比先验更可能成立——它已纳入证据信息。然而，当试图从多个证据/数据中学习时，情况就不那么清晰了。为此存在（至少）两种更新形式，分别与杰弗里和珀尔相关联。二者之间的差异往往未被明确识别。本文在离散概率论框架下提供导论与概述。文章始于一个涉及多项证据的基础性问题——该问题曾被发送给小型学术专家组。专家们的回答显示出显著差异。这被用作动机与起点，进而引入杰弗里与珀尔针对多输入提出的两种更新形式，并详细阐述其性质。最终，本文将此论述与认知理论中的所谓变分自由能（VFE）更新方法相联系。研究表明，杰弗里与珀尔方法均优于VFE更新，且VFE更新未必能如预期般减少散度（即纠正误差）。