Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Prediction-based uncertainty quantification for exchangeable sequences

Sandra Fortini

Sandra Fortini

Department of Decision Sciences, Bocconi University, via Roentgen 1, 20136 Milano, Italy

Contribution: Conceptualization, Formal analysis, Investigation, Methodology, Writing – original draft, Writing – review & editing

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Sonia Petrone

Sonia Petrone

Department of Decision Sciences, Bocconi University, via Roentgen 1, 20136 Milano, Italy

[email protected]

Contribution: Conceptualization, Formal analysis, Investigation, Methodology, Writing – original draft, Writing – review & editing

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    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’.

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    One contribution of 16 to a theme issue ‘Bayesian inference: challenges, perspectives, and prospects’.

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