arXiv subjects

Badr-Eddine Chérief-Abdellatif

Publications and source records attributed to Badr-Eddine Chérief-Abdellatif.

16 recordsLinked to original sources

arXiv2026

Parametric MMD Estimation with Missing Values: Robustness to Missingness and Data Model Misspecification

In the missing data literature, the Maximum Likelihood Estimator (MLE) is celebrated for its ignorability property under missing at random (MAR) data. However, its sensitivity to misspecification of the (complete) data model, even under MAR, remains a significant limitation. This issue is further exacerbated by the fact that the MAR assumption may not always be realistic, introducing an additional source of potential misspecification through the missingness mechanism. To address this, we propose a novel M-estimation procedure based on the Maximum Mean Discrepancy (MMD), which is provably robust to both model misspecification and deviations from the assumed missingness mechanism. Our approach offers strong theoretical guarantees and improved reliability in complex settings. We establish the consistency and asymptotic normality of the estimator under missing completely at random (MCAR), provide an efficient stochastic gradient descent algorithm, and derive error bounds that explicitly separate the contributions of model misspecification and missingness bias. Furthermore, we analyze missing not at random (MNAR) scenarios where our estimator maintains controlled error, including a Huber setting where both the missingness mechanism and the data model are contaminated. Our contributions refine the understanding of the limitations of the MLE and provide a robust and principled alternative for handling missing data.

stat.ME↗

arXiv2026

Convex Order Comparisons for Sub-Gamma Random Variables

Recent work has shown that sub-Gaussian random variables are dominated in convex order by a sharp multiple of a Gaussian. We study the analogous question for the sub-Gamma class underlying Bernstein's inequality, with the Laplace law as the majorant. Here the moment generating function is controlled by a Bernstein-type bound over a finite range of frequencies, rather than by a purely quadratic bound. We derive a variational formula for the optimal multiple, show that it is strictly larger than the natural scale $σ\vee α$, and prove that this value is sharp, being attained by an asymmetric two-point distribution. We then turn to the sub-exponential class, which has a quadratic bound as in the sub-Gaussian case, but only over a bounded range as in the sub-Gamma case. Interestingly, the sub-exponential class displays a different behavior: the optimal multiple is exactly $σ\vee α$, but is attained only when $α\leq σ$; when $α> σ$, the constant remains sharp, but equality cannot hold for any non-affine convex function. Both results rely on a sharp finite-range variant of the Kearns-Saul inequality, which is of independent interest.

math.PR↗

arXiv2026

Asymptotics of Nonparametric Estimation under general non-monotone MAR missingness: A Bayesian Approach

Missing values are ubiquitous in statistical practice, with potentially detrimental consequences for any statistical analysis. As such, a wealth of methods and theoretical results have been developed in the last decades. However, many questions remain open, in particular in the case of general non-monotone missing at random (MAR), where nonparametric results are still lacking. In this paper, we extend nonparametric Bayesian theory to this MAR setting. We introduce a general theorem of posterior contraction under MAR and an additional positivity condition and apply this result to density estimation as well as regression problems. In particular, we show that, despite the missing values, the complete-data density can be estimated with the minimax posterior contraction rate up to logarithmic factors. To the best of our knowledge, this is the first nonparametric result showing that the complete-data distribution can be consistently estimated under Rubin's MAR definition. As a consequence, we obtain an algorithm that takes incomplete data and returns a sample from a consistent estimate of the complete-data distribution.

math.ST↗

arXiv2026

From Bayes' Rule to Bayes Rules: Optimal Information Processing and Axiomatic Foundations Beyond Probability

This paper develops principled updating rules for possibilistic inference, where uncertainty about a fixed parameter is represented by a possibility function, the maxitive analogue of a probability distribution, and comparisons are made pointwise via a partial order. From two complementary foundations, an information-conservation viewpoint and an axiomatic viewpoint, we derive the same canonical update: the posterior is the prior-likelihood product followed by supremum normalisation. The two derivations agree for an arbitrary loss, differing only in where the learning-rate parameter enters. This parameter controls epistemic strength and is not identifiable from the normalising evidence alone, clarifying the role of analogous learning-rate parameters in generalised Bayesian updating.

math.ST↗

arXiv2026

Maxitive Donsker-Varadhan Formulation for Possibilistic Variational Inference

Variational inference (VI) is a cornerstone of modern Bayesian learning, enabling approximate inference in complex models. However, its formulation depends on expectations and divergences defined through high-dimensional integrals, often rendering analytical treatment impossible and necessitating heavy reliance on approximations. Possibility theory, an imprecise probability framework, allows us to directly model epistemic uncertainty instead of relying on a subjective interpretation of probabilities. While this framework provides robustness and interpretability under sparse or imprecise information, adapting VI to the possibilistic setting requires rethinking core concepts such as divergences, which presuppose additivity. In this work, we develop a principled formulation for performing possibilistic VI by establishing a maxitive analogue of the classical Donsker-Varadhan formulation. The resulting framework enables us to derive a learning rule for possibilistic VI with exponential-family candidates and practical update rules for neural-network training, giving rise to a family of optimizers termed CBOpt. Finally, we demonstrate that CBOpt achieves competitive performance on both in-domain and out-of-domain image classification tasks.

stat.ML↗

arXiv2025

Recursive PAC-Bayes: A Frequentist Approach to Sequential Prior Updates with No Information Loss

PAC-Bayesian analysis is a frequentist framework for incorporating prior knowledge into learning. It was inspired by Bayesian learning, which allows sequential data processing and naturally turns posteriors from one processing step into priors for the next. However, despite two and a half decades of research, the ability to update priors sequentially without losing confidence information along the way remained elusive for PAC-Bayes. While PAC-Bayes allows construction of data-informed priors, the final confidence intervals depend only on the number of points that were not used for the construction of the prior, whereas confidence information in the prior, which is related to the number of points used to construct the prior, is lost. This limits the possibility and benefit of sequential prior updates, because the final bounds depend only on the size of the final batch. We present a novel and, in retrospect, surprisingly simple and powerful PAC-Bayesian procedure that allows sequential prior updates with no information loss. The procedure is based on a novel decomposition of the expected loss of randomized classifiers. The decomposition rewrites the loss of the posterior as an excess loss relative to a downscaled loss of the prior plus the downscaled loss of the prior, which is bounded recursively. As a side result, we also present a generalization of the split-kl and PAC-Bayes-split-kl inequalities to discrete random variables, which we use for bounding the excess losses, and which can be of independent interest. In empirical evaluation the new procedure significantly outperforms state-of-the-art.

cs.LG↗

arXiv2025

Finite sample properties of parametric MMD estimation: robustness to misspecification and dependence

Many works in statistics aim at designing a universal estimation procedure, that is, an estimator that would converge to the best approximation of the (unknown) data generating distribution in a model, without any assumption on this distribution. This question is of major interest, in particular because the universality property leads to the robustness of the estimator. In this paper, we tackle the problem of universal estimation using a minimum distance estimator presented in Briol et al. (2019) based on the Maximum Mean Discrepancy. We show that the estimator is robust to both dependence and to the presence of outliers in the dataset. Finally, we provide a theoretical study of the stochastic gradient descent algorithm used to compute the estimator, and we support our findings with numerical simulations. ** The proof of Proposition 4.4 in the published version contains a mistake. The mistake is fixed here (and the bound is actually improved by a factor 2). **

math.ST↗

arXiv2023

Label Shift Quantification with Robustness Guarantees via Distribution Feature Matching

Quantification learning deals with the task of estimating the target label distribution under label shift. In this paper, we first present a unifying framework, distribution feature matching (DFM), that recovers as particular instances various estimators introduced in previous literature. We derive a general performance bound for DFM procedures, improving in several key aspects upon previous bounds derived in particular cases. We then extend this analysis to study robustness of DFM procedures in the misspecified setting under departure from the exact label shift hypothesis, in particular in the case of contamination of the target by an unknown distribution. These theoretical findings are confirmed by a detailed numerical study on simulated and real-world datasets. We also introduce an efficient, scalable and robust version of kernel-based DFM using the Random Fourier Feature principle.

stat.ML↗

arXiv2023

Bayes meets Bernstein at the Meta Level: an Analysis of Fast Rates in Meta-Learning with PAC-Bayes

Bernstein's condition is a key assumption that guarantees fast rates in machine learning. For example, the Gibbs algorithm with prior $π$ has an excess risk in $O(d_π/n)$, as opposed to the standard $O(\sqrt{d_π/n})$, where $n$ denotes the number of observations and $d_π$ is a complexity parameter which depends on the prior $π$. In this paper, we examine the Gibbs algorithm in the context of meta-learning, i.e., when learning the prior $π$ from $T$ tasks (with $n$ observations each) generated by a meta distribution. Our main result is that Bernstein's condition always holds at the meta level, regardless of its validity at the observation level. This implies that the additional cost to learn the Gibbs prior $π$, which will reduce the term $d_π$ across tasks, is in $O(1/T)$, instead of the expected $O(1/\sqrt{T})$. We further illustrate how this result improves on standard rates in three different settings: discrete priors, Gaussian priors and mixture of Gaussians priors.

stat.ML↗

arXiv2022

On PAC-Bayesian reconstruction guarantees for VAEs

Despite its wide use and empirical successes, the theoretical understanding and study of the behaviour and performance of the variational autoencoder (VAE) have only emerged in the past few years. We contribute to this recent line of work by analysing the VAE's reconstruction ability for unseen test data, leveraging arguments from the PAC-Bayes theory. We provide generalisation bounds on the theoretical reconstruction error, and provide insights on the regularisation effect of VAE objectives. We illustrate our theoretical results with supporting experiments on classical benchmark datasets.

cs.LG↗

arXiv2022

Estimation of copulas via Maximum Mean Discrepancy

This paper deals with robust inference for parametric copula models. Estimation using Canonical Maximum Likelihood might be unstable, especially in the presence of outliers. We propose to use a procedure based on the Maximum Mean Discrepancy (MMD) principle. We derive non-asymptotic oracle inequalities, consistency and asymptotic normality of this new estimator. In particular, the oracle inequality holds without any assumption on the copula family, and can be applied in the presence of outliers or under misspecification. Moreover, in our MMD framework, the statistical inference of copula models for which there exists no density with respect to the Lebesgue measure on $[0,1]^d$, as the Marshall-Olkin copula, becomes feasible. A simulation study shows the robustness of our new procedures, especially compared to pseudo-maximum likelihood estimation. An R package implementing the MMD estimator for copula models is available.

stat.ME↗

arXiv2019

MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy

In some misspecified settings, the posterior distribution in Bayesian statistics may lead to inconsistent estimates. To fix this issue, it has been suggested to replace the likelihood by a pseudo-likelihood, that is the exponential of a loss function enjoying suitable robustness properties. In this paper, we build a pseudo-likelihood based on the Maximum Mean Discrepancy, defined via an embedding of probability distributions into a reproducing kernel Hilbert space. We show that this MMD-Bayes posterior is consistent and robust to model misspecification. As the posterior obtained in this way might be intractable, we also prove that reasonable variational approximations of this posterior enjoy the same properties. We provide details on a stochastic gradient algorithm to compute these variational approximations. Numerical simulations indeed suggest that our estimator is more robust to misspecification than the ones based on the likelihood.

math.ST↗

arXiv2019

A Generalization Bound for Online Variational Inference

Bayesian inference provides an attractive online-learning framework to analyze sequential data, and offers generalization guarantees which hold even with model mismatch and adversaries. Unfortunately, exact Bayesian inference is rarely feasible in practice and approximation methods are usually employed, but do such methods preserve the generalization properties of Bayesian inference ? In this paper, we show that this is indeed the case for some variational inference (VI) algorithms. We consider a few existing online, tempered VI algorithms, as well as a new algorithm, and derive their generalization bounds. Our theoretical result relies on the convexity of the variational objective, but we argue that the result should hold more generally and present empirical evidence in support of this. Our work in this paper presents theoretical justifications in favor of online algorithms relying on approximate Bayesian methods.

stat.ML↗

arXiv2019

Convergence Rates of Variational Inference in Sparse Deep Learning

Variational inference is becoming more and more popular for approximating intractable posterior distributions in Bayesian statistics and machine learning. Meanwhile, a few recent works have provided theoretical justification and new insights on deep neural networks for estimating smooth functions in usual settings such as nonparametric regression. In this paper, we show that variational inference for sparse deep learning retains the same generalization properties than exact Bayesian inference. In particular, we highlight the connection between estimation and approximation theories via the classical bias-variance trade-off and show that it leads to near-minimax rates of convergence for Hölder smooth functions. Additionally, we show that the model selection framework over the neural network architecture via ELBO maximization does not overfit and adaptively achieves the optimal rate of convergence.

math.ST↗

arXiv2019

Consistency of ELBO maximization for model selection

The Evidence Lower Bound (ELBO) is a quantity that plays a key role in variational inference. It can also be used as a criterion in model selection. However, though extremely popular in practice in the variational Bayes community, there has never been a general theoretic justification for selecting based on the ELBO. In this paper, we show that the ELBO maximization strategy has strong theoretical guarantees, and is robust to model misspecification while most works rely on the assumption that one model is correctly specified. We illustrate our theoretical results by an application to the selection of the number of principal components in probabilistic PCA.

math.ST↗

arXiv2018

Consistency of Variational Bayes Inference for Estimation and Model Selection in Mixtures

Mixture models are widely used in Bayesian statistics and machine learning, in particular in computational biology, natural language processing and many other fields. Variational inference, a technique for approximating intractable posteriors thanks to optimization algorithms, is extremely popular in practice when dealing with complex models such as mixtures. The contribution of this paper is two-fold. First, we study the concentration of variational approximations of posteriors, which is still an open problem for general mixtures, and we derive consistency and rates of convergence. We also tackle the problem of model selection for the number of components: we study the approach already used in practice, which consists in maximizing a numerical criterion (the Evidence Lower Bound). We prove that this strategy indeed leads to strong oracle inequalities. We illustrate our theoretical results by applications to Gaussian and multinomial mixtures.

math.ST↗