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Ismaël Castillo

Publications and source records attributed to Ismaël Castillo.

At least 19 recordsLinked to original sources
arXiv2026

A variational Bayes approach to inference for low-dimensional parameters in high-dimensional linear regression

We propose a scalable variational Bayes method for statistical inference for a single or pre-specified low-dimensional subset of the coordinates of a high-dimensional parameter in sparse linear regression. Our approach relies on assigning a mean-field approximation to the nuisance coordinates and carefully modelling the conditional distribution of the target given the nuisance. This requires only a preprocessing step and preserves the computational advantages of mean-field variational Bayes, while ensuring accurate and reliable inference for the target parameter, including for uncertainty quantification. We investigate the numerical performance of our algorithm, showing that it performs competitively with existing methods. We further establish accompanying theoretical guarantees for estimation and uncertainty quantification in the form of a Bernstein--von Mises theorem for a possibly growing subset.

stat.ML↗
arXiv2026

Multiple testing with the horseshoe

We study multiple testing under continuous global--local shrinkage priors, with a focus on the horseshoe prior in high-dimensional sparse settings. While such priors provide adaptive shrinkage and computational scalability, they do not induce exact zeros and hence do not directly yield posterior inclusion probabilities, making principled false discovery control nontrivial. We propose posterior--based decision rules for signal detection that are applicable across a broad class of continuous shrinkage priors and are calibrated to control the false discovery rate (FDR) while retaining high power. In the sparse normal means model, we show that the proposed procedures attain the optimal detection boundary and achieve frequentist asymptotic control of both FDR and false negative rate (FNR). The method is readily implementable via standard posterior sampling, and empirical studies indicate that the realised FDR and FNR closely track their theoretical targets. Applications to high-dimensional regression and Gaussian graphical models further illustrate the scope and practical effectiveness of the approach.

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arXiv2026

Bayesian inference in high-dimensional models

Models with dimension more than the available sample size are now commonly used in various applications. A sensible inference is possible using a lower-dimensional structure. In regression problems with a large number of predictors, the model is often assumed to be sparse, with only a few predictors active. Interdependence between a large number of variables is succinctly described by a graphical model, where variables are represented by nodes on a graph and an edge between two nodes is used to indicate their conditional dependence given other variables. Many procedures for making inferences in the high-dimensional setting, typically using penalty functions to induce sparsity in the solution obtained by minimizing a loss function, were developed. Bayesian methods have been proposed for such problems more recently, where the prior takes care of the sparsity structure. These methods have the natural ability to also automatically quantify the uncertainty of the inference through the posterior distribution. Theoretical studies of Bayesian procedures in high-dimension have been carried out recently. Questions that arise are, whether the posterior distribution contracts near the true value of the parameter at the minimax optimal rate, whether the correct lower-dimensional structure is discovered with high posterior probability, and whether a credible region has adequate frequentist coverage. In this paper, we review these properties of Bayesian and related methods for several high-dimensional models such as many normal means problem, linear regression, generalized linear models, Gaussian and non-Gaussian graphical models. Effective computational approaches are also discussed.

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arXiv2026

Leveraging tails for adaptation

We consider contraction of Bayesian posterior distributions in nonparametric settings where coefficients of a function over a basis or dictionary are given priors with $p$--exponential tails, including Laplace tails $(p=1)$ and heavier tails $(p<1)$. It is shown that contraction rates improve as $p$ decreases and that full adaptation to smoothness, up to logarithmic factors, is obtained in an appropriate $p\to 0$ regime. As applications, we consider both series priors in white noise regression and shallow ReLU neural networks in random design regression. In particular, we show that overparametrised shallow ReLU networks can adapt to any regularity $0\le β\le 2$. Through a simulation study, we show strong empirical agreement with the behavior predicted by our theory.

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arXiv2026

Heavy-tailed and Horseshoe priors for regression and sparse Besov rates

The large variety of functions encountered in nonparametric statistics, calls for methods that are flexible enough to achieve optimal or near-optimal performance over a wide variety of functional classes, such as Besov balls, as well as over a large array of loss functions. In this work, we show that a class of heavy-tailed prior distributions on basis function coefficients introduced in \cite{AC} and called Oversmoothed heavy-Tailed (OT) priors, leads to Bayesian posterior distributions that satisfy these requirements; the case of horseshoe distributions is also investigated, for the first time in the context of nonparametrics, and we show that they fit into this framework. Posterior contraction rates are derived in two settings. The case of Sobolev--smooth signals and $L_2$--risk is considered first, along with a lower bound result showing that the imposed form of the scalings on prior coefficients by the OT prior is necessary to get full adaptation to smoothness. Second, the broader case of Besov-smooth signals with $L_{p'}$--risks, $p' \geq 1$, is considered, and minimax posterior contraction rates, adaptive to the underlying smoothness, and including rates in the so-called {\em sparse} zone, are derived. We provide an implementation of the proposed method and illustrate our results through a simulation study.

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arXiv2025

Posterior and variational inference for deep neural networks with heavy-tailed weights

We consider deep neural networks in a Bayesian framework with a prior distribution sampling the network weights at random. Following a recent idea of Agapiou and Castillo (2023), who show that heavy-tailed prior distributions achieve automatic adaptation to smoothness, we introduce a simple Bayesian deep learning prior based on heavy-tailed weights and ReLU activation. We show that the corresponding posterior distribution achieves near-optimal minimax contraction rates, simultaneously adaptive to both intrinsic dimension and smoothness of the underlying function, in a variety of contexts including nonparametric regression, geometric data and Besov spaces. While most works so far need a form of model selection built-in within the prior distribution, a key aspect of our approach is that it does not require to sample hyperparameters to learn the architecture of the network. We also provide variational Bayes counterparts of the results, that show that mean-field variational approximations still benefit from near-optimal theoretical support.

stat.ML↗
arXiv2025

Deep Horseshoe Gaussian Processes

Deep Gaussian processes have recently been proposed as natural objects to fit, similarly to deep neural networks, possibly complex features present in modern data samples, such as compositional structures. Adopting a Bayesian nonparametric approach, it is natural to use deep Gaussian processes as prior distributions, and use the corresponding posterior distributions for statistical inference. We introduce the deep Horseshoe Gaussian process Deep-HGP, a new simple prior based on deep Gaussian processes with a squared-exponential kernel, that in particular enables data-driven choices of the key lengthscale parameters. For nonparametric regression with random design, we show that the associated posterior distribution recovers the unknown true regression curve optimally in terms of quadratic loss, up to a logarithmic factor, in an adaptive way. The convergence rates are simultaneously adaptive to both the smoothness of the regression function and to its structure in terms of compositions. The dependence of the rates in terms of dimension are explicit, allowing in particular for input spaces of dimension increasing with the number of observations.

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arXiv2024

Heavy-tailed Bayesian nonparametric adaptation

We propose a new Bayesian strategy for adaptation to smoothness in nonparametric models based on heavy tailed series priors. We illustrate it in a variety of settings, showing in particular that the corresponding Bayesian posterior distributions achieve adaptive rates of contraction in the minimax sense (up to logarithmic factors) without the need to sample hyperparameters. Unlike many existing procedures, where a form of direct model (or estimator) selection is performed, the method can be seen as performing a soft selection through the prior tail. In Gaussian regression, such heavy tailed priors are shown to lead to (near-)optimal simultaneous adaptation both in the $L^2$- and $L^\infty$-sense. Results are also derived for linear inverse problems, for anisotropic Besov classes, and for certain losses in more general models through the use of tempered posterior distributions. We present numerical simulations corroborating the theory.

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arXiv2024

Semiparametric inference using fractional posteriors

We establish a general Bernstein--von Mises theorem for approximately linear semiparametric functionals of fractional posterior distributions based on nonparametric priors. This is illustrated in a number of nonparametric settings and for different classes of prior distributions, including Gaussian process priors. We show that fractional posterior credible sets can provide reliable semiparametric uncertainty quantification, but have inflated size. To remedy this, we further propose a \textit{shifted-and-rescaled} fractional posterior set that is an efficient confidence set having optimal size under regularity conditions. As part of our proofs, we also refine existing contraction rate results for fractional posteriors by sharpening the dependence of the rate on the fractional exponent.

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arXiv2023

Bayesian Multiscale Analysis of the Cox Model

Piecewise constant priors are routinely used in the Bayesian Cox proportional hazards model for survival analysis. Despite its popularity, large sample properties of this Bayesian method are not yet well understood. This work provides a unified theory for posterior distributions in this setting, not requiring the priors to be conjugate. We first derive contraction rate results for wide classes of histogram priors on the unknown hazard function and prove asymptotic normality of linear functionals of the posterior hazard in the form of Bernstein--von Mises theorems. Second, using recently developed multiscale techniques, we derive functional limiting results for the cumulative hazard and survival function. Frequentist coverage properties of Bayesian credible sets are investigated: we prove that certain easily computable credible bands for the survival function are optimal frequentist confidence bands. We conduct simulation studies that confirm these predictions, with an excellent behavior particularly in finite samples. Our results suggest that the Bayesian approach can provide an easy solution to obtain both the coefficients estimate and the credible bands for survival function in practice.

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arXiv2022

Uniform estimation in stochastic block models is slow

We explicitly quantify the empirically observed phenomenon that estimation under a stochastic block model (SBM) is hard if the model contains classes that are similar. More precisely, we consider estimation of certain functionals of random graphs generated by a SBM. The SBM may or may not be sparse, and the number of classes may be fixed or grow with the number of vertices. Minimax lower and upper bounds of estimation along specific submodels are derived. The results are nonasymptotic and imply that uniform estimation of a single connectivity parameter is much slower than the expected asymptotic pointwise rate. Specifically, the uniform quadratic rate does not scale as the number of edges, but only as the number of vertices. The lower bounds are local around any possible SBM. An analogous result is derived for functionals of a class of smooth graphons.

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arXiv2021

Optional Pólya trees: posterior rates and uncertainty quantification

We consider statistical inference in the density estimation model using a tree-based Bayesian approach, with Optional Pólya trees as prior distribution. We derive near-optimal convergence rates for corresponding posterior distributions with respect to the supremum norm. For broad classes of Hölder-smooth densities, we show that the method automatically adapts to the unknown Hölder regularity parameter. We consider the question of uncertainty quantification by providing mathematical guarantees for credible sets from the obtained posterior distributions, leading to near-optimal uncertainty quantification for the density function, as well as related functionals such as the cumulative distribution function. The results are illustrated through a brief simulation study.

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arXiv2021

Multiscale Bayesian Survival Analysis

We consider Bayesian nonparametric inference in the right-censoring survival model, where modeling is made at the level of the hazard rate. We derive posterior limiting distributions for linear functionals of the hazard, and then for `many' functionals simultaneously in appropriate multiscale spaces. As an application, we derive Bernstein-von Mises theorems for the cumulative hazard and survival functions, which lead to asymptotically efficient confidence bands for these quantities. Further, we show optimal posterior contraction rates for the hazard in terms of the supremum norm. In medical studies, a popular approach is to model hazards a priori as random histograms with possibly dependent heights. This and more general classes of arbitrarily smooth prior distributions are considered as applications of our theory. A sampler is provided for possibly dependent histogram posteriors. Its finite sample properties are investigated on both simulated and real data experiments.

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arXiv2020

Spike and Slab Pólya tree posterior distributions: adaptive inference

In the density estimation model, the question of adaptive inference using Pólya tree-type prior distributions is considered. A class of prior densities having a tree structure, called spike-and-slab Pólya trees, is introduced. For this class, two types of results are obtained: first, the Bayesian posterior distribution is shown to converge at the minimax rate for the supremum norm in an adaptive way, for any Hölder regularity of the true density between $0$ and $1$, thereby providing adaptive counterparts to the results for classical Pólya trees in Castillo (2017). Second, the question of uncertainty quantification is considered. An adaptive nonparametric Bernstein-von Mises theorem is derived. Next, it is shown that, under a self-similarity condition on the true density, certain credible sets from the posterior distribution are adaptive confidence bands, having prescribed coverage level and with a diameter shrinking at optimal rate in the minimax sense.

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arXiv2018

Empirical Bayes analysis of spike and slab posterior distributions

In the sparse normal means model, convergence of the Bayesian posterior distribution associated to spike and slab prior distributions is considered. The key sparsity hyperparameter is calibrated via marginal maximum likelihood empirical Bayes. The plug-in posterior squared-$L^2$ norm is shown to converge at the minimax rate for the euclidean norm for appropriate choices of spike and slab distributions. Possible choices include standard spike and slab with heavy tailed slab, and the spike and slab LASSO of Rocková and George with heavy tailed slab. Surprisingly, the popular Laplace slab is shown to lead to a suboptimal rate for the full empirical Bayes posterior. This provides a striking example where convergence of aspects of the empirical Bayes posterior does not entail convergence of the full empirical Bayes posterior itself.

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arXiv2015

A Bernstein-von Mises theorem for smooth functionals in semiparametric models

A Bernstein-von Mises theorem is derived for general semiparametric functionals. The result is applied to a variety of semiparametric problems in i.i.d. and non-i.i.d. situations. In particular, new tools are developed to handle semiparametric bias, in particular for nonlinear functionals and in cases where regularity is possibly low. Examples include the squared $L^2$-norm in Gaussian white noise, nonlinear functionals in density estimation, as well as functionals in autoregressive models. For density estimation, a systematic study of BvM results for two important classes of priors is provided, namely random histograms and Gaussian process priors.

math.ST↗
arXiv2015

Bayesian linear regression with sparse priors

We study full Bayesian procedures for high-dimensional linear regression under sparsity constraints. The prior is a mixture of point masses at zero and continuous distributions. Under compatibility conditions on the design matrix, the posterior distribution is shown to contract at the optimal rate for recovery of the unknown sparse vector, and to give optimal prediction of the response vector. It is also shown to select the correct sparse model, or at least the coefficients that are significantly different from zero. The asymptotic shape of the posterior distribution is characterized and employed to the construction and study of credible sets for uncertainty quantification.

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