arXiv subjects

Mylène Bédard

Publications and source records attributed to Mylène Bédard.

7 recordsLinked to original sources

arXiv2026

Adaptive Pseudo-Marginal Algorithm

The Pseudo-Marginal (PM) algorithm is a popular Markov chain Monte Carlo (MCMC) method used to sample from a target distribution when its density is inaccessible, but can be estimated with a non-negative unbiased estimator. Its performance depends on a key parameter, N, the number of iterations (or particles) used to approximate the target density. Larger values of N yield more accurate estimates but at increased running time. Previous studies has provided guidelines for selecting an optimal value of N to balance this tradeoff. However, this approach involves multiple steps and manual adjustments. To overcome these limitations, we introduce an adaptive version of the PM algorithm, where N is automatically adjusted during the iterative process toward its optimal value, thus eliminating the need for manual intervention. This algorithm ensures convergence under certain conditions. On two examples, including a real data problem on pulmonary infection in preschool children, the proposed algorithm compares favorably to the existing approach.

stat.CO↗

arXiv2020

The Linear Lasso: a location model resolution

We use location model methodology to guide the least squares analysis of the Lasso problem of variable selection and inference. The nuisance parameter is taken to be an indicator for the selection of explanatory variables and the interest parameter is the response variable itself. Recent theory eliminates the nuisance parameter by marginalization on the data space and then uses the resulting distribution for inference concerning the interest parameter. We develop this approach and find: that primary inference is essentially one-dimensional rather than $n$-dimensional; that inference focuses on the response variable itself rather than the least squares estimate (as variables are removed); that first order probabilities are available; that computation is relatively easy; that a scalar marginal model is available; and that ineffective variables can be removed by distributional tilt or shift.

stat.ME↗

arXiv2020

An automatic robust Bayesian approach to principal component regression

Principal component regression uses principal components as regressors. It is particularly useful in prediction settings with high-dimensional covariates. The existing literature treating of Bayesian approaches is relatively sparse. We introduce a Bayesian approach that is robust to outliers in both the dependent variable and the covariates. Outliers can be thought of as observations that are not in line with the general trend. The proposed approach automatically penalises these observations so that their impact on the posterior gradually vanishes as they move further and further away from the general trend, corresponding to a concept in Bayesian statistics called whole robustness. The predictions produced are thus consistent with the bulk of the data. The approach also exploits the geometry of principal components to efficiently identify those that are significant. Individual predictions obtained from the resulting models are consolidated according to model-averaging mechanisms to account for model uncertainty. The approach is evaluated on real data and compared to its nonrobust Bayesian counterpart, the traditional frequentist approach, and a commonly employed robust frequentist method. Detailed guidelines to automate the entire statistical procedure are provided. All required code is made available, see ArXiv:1711.06341.

stat.ME↗

arXiv2019

A New Bayesian Approach to Robustness Against Outliers in Linear Regression

Linear regression is ubiquitous in statistical analysis. It is well understood that conflicting sources of information may contaminate the inference when the classical normality of errors is assumed. The contamination caused by the light normal tails follows from an undesirable effect: the posterior concentrates in an area in between the different sources with a large enough scaling to incorporate them all. The theory of conflict resolution in Bayesian statistics (O'Hagan and Pericchi (2012)) recommends to address this problem by limiting the impact of outliers to obtain conclusions consistent with the bulk of the data. In this paper, we propose a model with super heavy-tailed errors to achieve this. We prove that it is wholly robust, meaning that the impact of outliers gradually vanishes as they move further and further away form the general trend. The super heavy-tailed density is similar to the normal outside of the tails, which gives rise to an efficient estimation procedure. In addition, estimates are easily computed. This is highlighted via a detailed user guide, where all steps are explained through a simulated case study. The performance is shown using simulation. All required code is given.

stat.ME↗

arXiv2019

Weak Convergence and Optimal Tuning of the Reversible Jump Algorithm

The reversible jump algorithm is a useful Markov chain Monte Carlo method introduced by Green (1995) that allows switches between subspaces of differing dimensionality, and therefore, model selection. Although this method is now increasingly used in key areas (e.g. biology and finance), it remains a challenge to implement it. In this paper, we focus on a simple sampling context in order to obtain theoretical results that lead to an optimal tuning procedure for the considered reversible jump algorithm, and consequently, to easy implementation. The key result is the weak convergence of the sequence of stochastic processes engendered by the algorithm. It represents the main contribution of this paper as it is, to our knowledge, the first weak convergence result for the reversible jump algorithm. The sampler updating the parameters according to a random walk, this result allows to retrieve the well-known 0.234 rule for finding the optimal scaling. It also leads to an answer to the question: "with what probability should a parameter update be proposed comparatively to a model switch at each iteration?"

stat.ME↗

arXiv2017

A Dirichlet Form approach to MCMC Optimal Scaling

This paper develops the use of Dirichlet forms to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis-Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Dirichlet form methods have the added advantage of providing an explicit construction of the underlying infinite-dimensional context. In particular, this enables us directly to establish weak convergence to the relevant infinite-dimensional distributions.

math.PR↗

arXiv2007

Weak convergence of Metropolis algorithms for non-i.i.d. target distributions

In this paper, we shall optimize the efficiency of Metropolis algorithms for multidimensional target distributions with scaling terms possibly depending on the dimension. We propose a method for determining the appropriate form for the scaling of the proposal distribution as a function of the dimension, which leads to the proof of an asymptotic diffusion theorem. We show that when there does not exist any component with a scaling term significantly smaller than the others, the asymptotically optimal acceptance rate is the well-known 0.234.

math.PR↗