  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.
