Prediction-based uncertainty quantification for exchangeable sequences

Prediction has a central role in the foundations of Bayesian statistics and is now the main focus in many areas of machine learning, in contrast to the more classical focus on inference. We discuss that, in the basic setting of random sampling-that is, in the Bayesian approach, exchangeability-uncertainty expressed by the posterior distribution and credible intervals can indeed be understood in terms of prediction. The posterior law on the unknown distribution is centred on the predictive distribution and we prove that it is marginally asymptotically Gaussian with variance depending on the predictive updates, i.e. on how the predictive rule incorporates information as new observations become available. This allows to obtain asymptotic credible intervals only based on the predictive rule (without having to specify the model and the prior law), sheds light on frequentist coverage as related to the predictive learning rule, and, we believe, opens a new perspective towards a notion of predictive efficiency that seems to call for further research. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'.