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Abstract:
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We review asymptotic properties of posterior distributions on infinite-dimensional parameter spaces under the non-Bayesian assumption that the data is generated according to some fixed true distribution. For instance, we are interested in whether and how fast a sequence of posterior distributions contracts to the true distribution if the information in the data increases indefinitely, or in distributional properties of posterior marginal distribution (Bernstein-von Mises theorem). We review some general results, and their relation to earlier work by Lucien le Cam, and various examples. In particular, we may discuss using randomly rescaled Gaussian processes as priors to estimate smooth regression, classification or density functions, priors for sparsity, or adaptation by Bayesian methods.
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