Rates of contraction for posterior distributions in $\bolds{L^r}$-metrics, $\bolds{1\le r\le\infty}$

TL;DR

This paper investigates the rates at which Bayesian posterior distributions concentrate around the true parameter in various L^r-norms, providing theoretical results and optimal rates for different prior models in nonparametric density estimation and regression.

Contribution

It establishes general theorems for posterior contraction rates in L^r-norms, demonstrating minimax-optimal rates for 1≤r≤2 and devising a Gaussian prior achieving optimal contraction in all L^r-norms.

Findings

Rates are minimax-optimal for 1≤r≤2.

Rates deteriorate for r>2.

Gaussian prior achieves optimal rates in all L^r-norms for regression.

Abstract

The frequentist behavior of nonparametric Bayes estimates, more specifically, rates of contraction of the posterior distributions to shrinking $L^r$-norm neighborhoods, $1\le r\le\infty$, 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 $1\le r\le2$, 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, $1\le r\le\infty$.