Convergence rates for posterior distributions and adaptive estimation

Approximate Bayesian computation allows for statistical analysis in models with intractable likelihoods. In this paper we consider the asymptotic behaviour of the posterior distribution obtained by this method. We give general results on the rate at which the posterior distribution concentrates on sets containing the true parameter, its limiting shape, and the asymptotic distribution of the posterior mean. These results hold under given rates for the tolerance used within the...

We study location-scale mixture priors for nonparametric statistical problems, including multivariate regression, density estimation and classification. We show that a rate-adaptive procedure can be obtained if the prior is properly constructed. In particular, we show that adaptation is achieved if a kernel mixture prior on a regression function is constructed using a Gaussian kernel, an inverse gamma bandwidth, and Gaussian mixing weights.

In a Bayesian context, prior specification for inference on monotone densities is conceptually straightforward, but proving posterior convergence theorems is complicated by the fact that desirable prior concentration properties often are not satisfied. In this paper, I first develop a new prior designed specifically to satisfy an empirical version of the prior concentration property, and then I give sufficient conditions on the prior inputs such that the corresponding empiric...

We consider a nonparametric regression model $Y=r(X)+\varepsilon$ with a random covariate $X$ that is independent of the error $\varepsilon$. Then the density of the response $Y$ is a convolution of the densities of $\varepsilon$ and $r(X)$. It can therefore be estimated by a convolution of kernel estimators for these two densities, or more generally by a local von Mises statistic. If the regression function has a nowhere vanishing derivative, then the convolution estimator c...

Sup-norm curve estimation is a fundamental statistical problem and, in principle, a premise for the construction of confidence bands for infinite-dimensional parameters. In a Bayesian framework, the issue of whether the sup-norm-concentration- of-posterior-measure approach proposed by Gin\'e and Nickl (2011), which involves solving a testing problem exploiting concentration properties of kernel and projection-type density estimators around their expectations, can yield minima...

We study the rate of Bayesian consistency for hierarchical priors consisting of prior weights on a model index set and a prior on a density model for each choice of model index. Ghosal, Lember and Van der Vaart [2] have obtained general in-probability theorems on the rate of convergence of the resulting posterior distributions. We extend their results to almost sure assertions. As an application we study log spline densities with a finite number of models and obtain that the ...

In the context of density level set estimation, we study the convergence of general plug-in methods under two main assumptions on the density for a given level $\lambda$. More precisely, it is assumed that the density (i) is smooth in a neighborhood of $\lambda$ and (ii) has $\gamma$-exponent at level $\lambda$. Condition (i) ensures that the density can be estimated at a standard nonparametric rate and condition (ii) is similar to Tsybakov's margin assumption which is stated...

A scheme for locally adaptive bandwidth selection is proposed which sensitively shrinks the bandwidth of a kernel estimator at lowest density regions such as the support boundary which are unknown to the statistician. In case of a H\"{o}lder continuous density, this locally minimax-optimal bandwidth is shown to be smaller than the usual rate, even in case of homogeneous smoothness. Some new type of risk bound with respect to a density-dependent standardized loss of this estim...

In this paper we will consider the estimation of a monotone regression (or density) function in a fixed point by the least squares (Grenander) estimator. We will show that this estimator is fully adaptive, in the sense that the attained rate is given by a functional relation using the underlying function $f_0$, and not by some smoothness parameter, and that this rate is optimal when considering the class of all monotone functions, in the sense that there exists a sequence of ...

In this paper, we use the class of Wasserstein metrics to study asymptotic properties of posterior distributions. Our first goal is to provide sufficient conditions for posterior consistency. In addition to the well-known Schwartz's Kullback--Leibler condition on the prior, the true distribution and most probability measures in the support of the prior are required to possess moments up to an order which is determined by the order of the Wasserstein metric. We further investi...