Bt Botond Szabó

Empirical Bayes scaling of Gaussian priors in the white noise model

The performance of nonparametric estimators is heavily dependent on a bandwidth parameter. In nonparametric Bayesian methods this parameter can be specified as a hyperparameter of the nonparametric prior. The value of this hyperparameter may be made dependent on the data. The empirical Bayes method is to set its value by maximizing the marginal likelihood of the data in the Bayesian framework. In this paper we analyze a particular version of this method, common in practice, that the hyperparameter scales the prior variance. We characterize the behavior of the random hyperparameter, and show that a nonparametric Bayes method using it gives optimal recovery over a scale of regularity classes. This scale is limited, however, by the regularity of the unscaled prior. While a prior can be scaled up to make it appropriate for arbitrarily rough truths, scaling cannot increase the nominal smoothness by much. Surprisingy the standard empirical Bayes method is even more limited in this respect than an oracle, deterministic scaling method. The same can be said for the hierarchical Bayes method.

Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator

We consider the asymptotic behaviour of the marginal maximum likelihood empirical Bayes posterior distribution in general setting. First we characterize the set where the maximum marginal likelihood estimator is located with high probability. Then we provide oracle type of upper and lower bounds for the contraction rates of the empirical Bayes posterior. We also show that the hierarchical Bayes posterior achieves the same contraction rate as the maximum marginal likelihood empirical Bayes posterior. We demonstrate the applicability of our general results for various models and prior distributions by deriving upper and lower bounds for the contraction rates of the corresponding empirical and hierarchical Bayes posterior distributions.

Adaptive posterior contraction rates for the horseshoe

We investigate the frequentist properties of Bayesian procedures for estimation based on the horseshoe prior in the sparse multivariate normal means model. Previous theoretical results assumed that the sparsity level, that is, the number of signals, was known. We drop this assumption and characterize the behavior of the maximum marginal likelihood estimator (MMLE) of a key parameter of the horseshoe prior. We prove that the MMLE is an effective estimator of the sparsity level, in the sense that it leads to (near) minimax optimal estimation of the underlying mean vector generating the data. Besides this empirical Bayes procedure, we consider the hierarchical Bayes method of putting a prior on the unknown sparsity level as well. We show that both Bayesian techniques lead to rate-adaptive optimal posterior contraction, which implies that the horseshoe posterior is a good candidate for generating rate-adaptive credible sets.

Uncertainty Quantification for the Horseshoe (with Discussion)

We begin by introducing the main ideas of the paper under discussion. We discuss some interesting issues regarding adaptive component-wise credible intervals. We then briefly touch upon the concepts of self-similarity and excessive bias restriction. This is then followed by some comments on the extensive simulation study carried out in the paper.We begin by introducing the main ideas of the paper under discussion. We discuss some interesting issues regarding adaptive component-wise credible intervals. We then briefly touch upon the concepts of self-similarity and excessive bias restriction. This is then followed by some comments on the extensive simulation study carried out in the paper.

On Bayesian Based Adaptive Confidence Sets for Linear Functionals

We consider the problem of constructing Bayesian based confidence sets for linear functionals in the inverse Gaussian white noise model. We work with a scale of Gaussian priors indexed by a regularity hyper-parameter and apply the data-driven (slightly modified) marginal likelihood empirical Bayes method for the choice of this hyper-parameter. We show by theory and simulations that the credible sets constructed by this method have sub-optimal behaviour in general. However, by assuming “self-similarity” the credible sets have rate-adaptive size and optimal coverage. As an application of these results we construct L ∞ -credible bands for the true functional parameter with adaptive size and optimal coverage under self-similarity constraint.

Rejoinder to discussions of “Frequentist coverage of adaptive nonparametric Bayesian credible sets”

We thank the discussants for their supportive comments and interesting observations. Many questions are still open and not all methododlogical or philosophical questions may have an answer. Our reply adresses only a subset of questions and is organizaed by topic. A final section reviews recent work.

Uncertainty Quantification for the Horseshoe

We begin by introducing the main ideas of the paper under discussion. We discuss some interesting issues regarding adaptive component-wise credible intervals. We then briefly touch upon the concepts of self-similarity and excessive bias restriction. This is then followed by some comments on the extensive simulation study carried out in the paper.We begin by introducing the main ideas of the paper under discussion. We discuss some interesting issues regarding adaptive component-wise credible intervals. We then briefly touch upon the concepts of self-similarity and excessive bias restriction. This is then followed by some comments on the extensive simulation study carried out in the paper.