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Abstract:
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The celebrated Bernstein-von Mises theorem implies that for regular parametric problems, Bayesian credible sets are also approximately frequentist confidence sets. Thus the uncertainty quantification by the two approaches essentially agree even though they have very different interpretations. A frequentist can then construct confidence sets by Bayesian means, which are often easily obtained from posterior sampling. However the incredible agreement can fall apart in nonparametric problems, whenever the bias becomes prominent. Recently some positive results have appeared in the literature overcoming the problem by undersmoothing or inflation of credible sets. We shall discuss results on Bayes-frequentist agreement of uncertainty quantification in white noise models, nonparametric regression and high dimensional linear models. We shall also discuss some related results on nonlinear functionals.
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