arXiv subjects

Jean-Bernard Salomond

Publications and source records attributed to Jean-Bernard Salomond.

5 recordsLinked to original sources

arXiv2017

Risk quantification for the thresholding rule for multiple testing using Gaussian scale mixtures

In this paper we study the asymptotic properties of Bayesian multiple testing procedures for a large class of Gaussian scale mixture pri- ors. We study two types of multiple testing risks: a Bayesian risk proposed in Bogdan et al. (2011) where the data are assume to come from a mixture of normal, and a frequentist risk similar to the one proposed by Arias-Castro and Chen (2017). Following the work of van der Pas et al. (2016), we give general conditions on the prior such that both risks can be bounded. For the Bayesian risk, the bound is almost sharp. This result show that under these conditions, the considered class of continuous prior can be competitive with the usual two-group model (e.g. spike and slab priors). We also show that if the non-zeros component of the parameter are large enough, the minimax risk can be made asymptotically null. The separation rates obtained are consistent with the one that could be guessed from the existing literature (see van der Pas et al., 2017b). For both problems, we then give conditions under which an adaptive version of the result can be obtained.

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arXiv2017

Testing Un-Separated Hypotheses by Estimating a Distance

In this paper we propose a Bayesian answer to testing problems when the hypotheses are not well separated. The idea of the method is to study the posterior distribution of a discrepancy measure between the parameter and the model we want to test for. This is shown to be equivalent to a modification of the testing loss. An advantage of this approach is that it can easily be adapted to complex hypotheses testing which are in general difficult to test for. Asymptotic properties of the test can be derived from the asymptotic behaviour of the posterior distribution of the discrepancy measure, and gives insight on possible calibrations. In addition one can derive separation rates for testing, which ensure the asymptotic frequentist optimality of our procedures.

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arXiv2017

A general approach to posterior contraction in nonparametric inverse problems

In this paper we propose a general method to derive an upper bound for the contraction rate of the posterior distribution for nonparametric inverse problems. We present a general theorem that allows us to derive con- traction rates for the parameter of interest from contraction rates of the related direct problem of estimating transformed parameter of interest. An interesting aspect of this approach is that it allows us to derive con- traction rates for priors that are not related to the singular value decomposition of the operator. We apply our result to several examples of linear inverse problems, both in the white noise sequence model and the nonparametric regression model, using priors based on the singular value decomposition of the operator, location-mixture priors and splines prior, and recover minimax adaptive contraction rates.

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arXiv2015

Conditions for Posterior Contraction in the Sparse Normal Means Problem

The first Bayesian results for the sparse normal means problem were proven for spike-and-slab priors. However, these priors are less convenient from a computational point of view. In the meanwhile, a large number of continuous shrinkage priors has been proposed. Many of these shrinkage priors can be written as a scale mixture of normals, which makes them particularly easy to implement. We propose general conditions on the prior on the local variance in scale mixtures of normals, such that posterior contraction at the minimax rate is assured. The conditions require tails at least as heavy as Laplace, but not too heavy, and a large amount of mass around zero relative to the tails, more so as the sparsity increases. These conditions give some general guidelines for choosing a shrinkage prior for estimation under a nearly black sparsity assumption. We verify these conditions for the class of priors considered by Ghosh and Chakrabarti (2015), which includes the horseshoe and the normal-exponential gamma priors, and for the horseshoe+, the inverse-Gaussian prior, the normal-gamma prior, and the spike-and-slab Lasso, and thus extend the number of shrinkage priors which are known to lead to posterior contraction at the minimax estimation rate.

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arXiv2014

Concentration rate and consistency of the posterior under monotonicity constraints

In this paper, we consider the well known problem of estimating a density function under qualitative assumptions. More precisely, we estimate monotone non increasing densities in a Bayesian setting and derive concentration rate for the posterior distribution for a Dirichlet process and finite mixture prior. We prove that the posterior distribution based on both priors concentrates at the rate $(n/\log(n))^{-1/3}$, which is the minimax rate of estimation up to a \log(n)$ factor. We also study the behaviour of the posterior for the point-wise loss at any fixed point of the support the density and for the sup norm. We prove that the posterior is consistent for both losses.

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