On adaptive posterior concentration rates

We investigate the problem of deriving posterior concentration rates under different loss functions in nonparametric Bayes. We first provide a lower bound on posterior coverages of shrinking neighbourhoods that relates the metric or loss under which the shrinking neighbourhood is considered, and an intrinsic pre-metric linked to frequentist separation rates. In the Gaussian white noise model, we construct feasible priors based on a spike and slab procedure reminiscent of wavelet thresholding that achieve adaptive rates of contraction under $L^2$ or $L^{\infty}$ metrics when the underlying parameter belongs to a collection of H\"{o}lder balls and that moreover achieve our lower bound. We analyse the consequences in terms of asymptotic behaviour of posterior credible balls as well as frequentist minimax adaptive estimation. Our results are appended with an upper bound for the contraction rate under an arbitrary loss in a generic regular experiment. The upper bound is attained for certain sieve priors and enables to extend our results to density estimation.Comment: Published at http://dx.doi.org/10.1214/15-AOS1341 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

High-dimensional Gaussian model selection on a Gaussian design

We consider the problem of estimating the conditional mean of a real Gaussian variable \nolinebreak Y=\sum_{i=1}^p\nolinebreak\theta_iX_i+\nolinebreak \epsilon where the vector of the covariates $(X_i)_{1\leq i\leq p}$ follows a joint Gaussian distribution. This issue often occurs when one aims at estimating the graph or the distribution of a Gaussian graphical model. We introduce a general model selection procedure which is based on the minimization of a penalized least-squares type criterion. It handles a variety of problems such as ordered and complete variable selection, allows to incorporate some prior knowledge on the model and applies when the number of covariates $p$ is larger than the number of observations $n$. Moreover, it is shown to achieve a non-asymptotic oracle inequality independently of the correlation structure of the covariates. We also exhibit various minimax rates of estimation in the considered framework and hence derive adaptiveness properties of our procedure

Adaptive estimation of covariance matrices via Cholesky decomposition

This paper studies the estimation of a large covariance matrix. We introduce a novel procedure called ChoSelect based on the Cholesky factor of the inverse covariance. This method uses a dimension reduction strategy by selecting the pattern of zero of the Cholesky factor. Alternatively, ChoSelect can be interpreted as a graph estimation procedure for directed Gaussian graphical models. Our approach is particularly relevant when the variables under study have a natural ordering (e.g. time series) or more generally when the Cholesky factor is approximately sparse. ChoSelect achieves non-asymptotic oracle inequalities with respect to the Kullback-Leibler entropy. Moreover, it satisfies various adaptive properties from a minimax point of view. We also introduce and study a two-stage procedure that combines ChoSelect with the Lasso. This last method enables the practitioner to choose his own trade-off between statistical efficiency and computational complexity. Moreover, it is consistent under weaker assumptions than the Lasso. The practical performances of the different procedures are assessed on numerical examples

Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing

An important estimation problem that is closely related to large-scale multiple testing is that of estimating the null density and the proportion of nonnull effects. A few estimators have been introduced in the literature; however, several important problems, including the evaluation of the minimax rate of convergence and the construction of rate-optimal estimators, remain open. In this paper, we consider optimal estimation of the null density and the proportion of nonnull effects. Both minimax lower and upper bounds are derived. The lower bound is established by a two-point testing argument, where at the core is the novel construction of two least favorable marginal densities $f_1$ and $f_2$. The density $f_1$ is heavy tailed both in the spatial and frequency domains and $f_2$ is a perturbation of $f_1$ such that the characteristic functions associated with $f_1$ and $f_2$ match each other in low frequencies. The minimax upper bound is obtained by constructing estimators which rely on the empirical characteristic function and Fourier analysis. The estimator is shown to be minimax rate optimal. Compared to existing methods in the literature, the proposed procedure not only provides more precise estimates of the null density and the proportion of the nonnull effects, but also yields more accurate results when used inside some multiple testing procedures which aim at controlling the False Discovery Rate (FDR). The procedure is easy to implement and numerical results are given.Comment: Published in at http://dx.doi.org/10.1214/09-AOS696 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Analysing correlated noise on the surface code using adaptive decoding algorithms

Laboratory hardware is rapidly progressing towards a state where quantum error-correcting codes can be realised. As such, we must learn how to deal with the complex nature of the noise that may occur in real physical systems. Single qubit Pauli errors are commonly used to study the behaviour of error-correcting codes, but in general we might expect the environment to introduce correlated errors to a system. Given some knowledge of structures that errors commonly take, it may be possible to adapt the error-correction procedure to compensate for this noise, but performing full state tomography on a physical system to analyse this structure quickly becomes impossible as the size increases beyond a few qubits. Here we develop and test new methods to analyse blue a particular class of spatially correlated errors by making use of parametrised families of decoding algorithms. We demonstrate our method numerically using a diffusive noise model. We show that information can be learnt about the parameters of the noise model, and additionally that the logical error rates can be improved. We conclude by discussing how our method could be utilised in a practical setting blue and propose extensions of our work to study more general error models.Comment: 19 pages, 8 figures, comments welcome; v2 - minor typos corrected some references added; v3 - accepted to Quantu

Unequal Error Protection Querying Policies for the Noisy 20 Questions Problem

In this paper, we propose an open-loop unequal-error-protection querying policy based on superposition coding for the noisy 20 questions problem. In this problem, a player wishes to successively refine an estimate of the value of a continuous random variable by posing binary queries and receiving noisy responses. When the queries are designed non-adaptively as a single block and the noisy responses are modeled as the output of a binary symmetric channel the 20 questions problem can be mapped to an equivalent problem of channel coding with unequal error protection (UEP). A new non-adaptive querying strategy based on UEP superposition coding is introduced whose estimation error decreases with an exponential rate of convergence that is significantly better than that of the UEP repetition coding introduced by Variani et al. (2015). With the proposed querying strategy, the rate of exponential decrease in the number of queries matches the rate of a closed-loop adaptive scheme where queries are sequentially designed with the benefit of feedback. Furthermore, the achievable error exponent is significantly better than that of random block codes employing equal error protection.Comment: To appear in IEEE Transactions on Information Theor