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Rates of contraction for posterior distributions in \(L^{r}\)-metrics, \(1 \leq r \leq \infty\). (English) Zbl 1246.62095
Summary: The frequentist behavior of nonparametric Bayes estimates, more specifically, rates of contraction of the posterior distributions to shrinking \(L^{r}\)-norm neighborhoods, \(1 \leq r \leq \infty \), of the unknown parameter, are studied. A theorem for nonparametric density estimation is proved under general approximation-theoretic assumptions on the prior. The result is applied to a variety of common examples, including Gaussian process, wavelet series, normal mixture and histogram priors. The rates of contraction are minimax-optimal for \(1 \leq r \leq 2\), but deteriorate as \(r\) increases beyond 2. In the case of Gaussian nonparametric regression a Gaussian prior is devised for which the posterior contracts at the optimal rate in all \(L^{r}\)-norms, \(1 \leq r \leq \infty \).

MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62G10 Nonparametric hypothesis testing
62F15 Bayesian inference
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