The increasing use of machine learning in sensitive applications demands algorithms that simultaneously preserve data privacy and ensure fairness across potentially sensitive subpopulations. While privacy and fairness have each been extensively studied, their joint treatment remains poorly understood. This talk will begin with an introduction to the key notions of fairness in machine learning and differential privacy, highlighting their objectives and the challenges involved in combining them. I will then present our recent work, in which we explore how privacy-preserving mechanisms can be incorporated into fairness-aware pipelines. In particular, we propose a post-processing approach, and we evaluate its performance in terms of the trade-offs between accuracy, fairness, and privacy. The results illustrate that it is possible to balance these objectives effectively in practice.

Hawkes processes are a popular model for self-exciting phenomena, from earthquakes to finance. In this talk, I will first present them in a simple way, using a Poisson imbedding construction. I will then review what is known about their long-time behavior, through limit theorems for both linear and non-linear cases. The focus will be on three regimes that appear when the process has a long memory and the branching ratio gets close to or above one: the Nearly Unstable, the Weakly Critical, and the Supercritical Nearly Unstable Hawkes processes. These regimes have been studied qualitatively, but quantitative convergence results have been missing. I will explain how we obtain explicit convergence rates, relying on a coupling with a Brownian sheet, Fourier analysis, and a careful approximation of the absolute value function.

The human gut microbiome is a complex ecosystem whose composition is derived from high-throughput sequencing technologies, yielding hierarchical count data indexed by microbial taxa. These microbiome profiles offer major opportunities for understanding host-microbes interactions, yet they come with substantial statistical challenges: counts are discrete, sparse, and overdispersed; sequencing induces compositional constraints and high-dimensional profiles, while clinical applications often take place in heterogeneous and small cohorts, with imperfect labeling regarding the underlying biology.

In this thesis, we develop structured latent generative models for microbiome count data and apply them to clinical studies. Our methodological contributions notably rely on Poisson log-normal (PLN) models which provide a principled probabilistic framework tailored to multivariate counts, and on variational inference which enables scalable learning while leveraging data structure. First, we incorporate the hierarchical organization of microbial taxa by introducing a tree-based extension of PLN models and by establishing identifiability results that support principled interpretation. We then build on this framework to propose a model-based data augmentation strategy that enhances predictive performance across clinical tasks while preserving ecological coherence. Subsequently, we extend our latent generative viewpoint to longitudinal settings through a perturbation-aware independent component analysis model for temporal count data, joined with identifiability guarantees allowing principled interpretation of the inferred components and regimes. Finally, we consider two inflammatory bowel disease case studies to highlight the clinical constraints that shape microbiome analyses, and to illustrate how parts of our methodology can be leveraged for prognosis in a statistically challenging setting.

Overall, this thesis argues that latent generative modeling offers a compelling framework for microbiome analysis, enabling the incorporation of biological structure while connecting representation learning, preprocessing, and data augmentation within a single probabilistic perspective. In particular, interpretability in latent microbiome models hinges on identifiability, and introducing structural information turns probabilistic models into principled tools for extracting meaningful representations and enhancing the statistical power of microbiome profiles.

Variational Inference (VI) is now a well established approach in statistical learning. It gave raise to many variants and algorithms used to optimize the variational parameter or the model parameter. The common first ingredient of these variants is to exhibit a convenient lower bound of the model likelihood. We will first present recent works on the inference of the variational parameter based on the monotonic optimization of the alpha divergence. However, leveraging reparameterization strategies, many algorithms are based on stochastic gradient descent, in which case controlling the bias and variance of the stochastic gradient is of particular interest. Several variational bounds involving importance weighting ideas have been proposed to generalize and improve on the Evidence Lower BOund (ELBO) in the context of marginal likelihood optimization, such as the Importance-weighted Auto-Encoder (IWAE), Variational Rényi (VR) and VR-IWAE bounds. Yet, it remains unclear how the joint choice of bound and gradient estimator impacts the behavior of the resulting algorithms. In this context, we will present some recent works studying reparameterized and doubly-reparameterized gradient estimators tied to the IWAE, VR and VR-IWAE bounds.

Dans un travail en collaboration avec Juhan Aru, nous considérons deux modèles naturels de fonctions de hauteur aléatoires sur des arbres avec conditions de bord zéro sur un sous-ensemble de feuilles ; à savoir, le champ libre gaussien discret, et un modèle analogue de fonctions de hauteur aléatoires à valeurs entières. Pour ces deux modèles, nous considérons la loi de la hauteur à la racine lorsque l'arbre sous-jacent consiste en les $n$ premières générations d'un arbre de Bienaymé surcritique, et chaque feuille est gardée dans le bord indépendamment avec probabilité $p_n$. Une transition de phase localisation/délocalisation apparaît alors autour de $p_n=n/m^n$, où $m>1$ est l'espérance de la loi de reproduction, et nous établissons une limite d'échelle gaussienne pour la hauteur à la racine dans le modèle à valeurs entières dans le régime délocalisé. Notre preuve repose sur un résultat de concentration quenched pour les marches branchantes surcritiques.

Résumé: Cette thèse étudie le comportement transient de certains processus de diffusion planaires avec interactions au bord, telles que des réflexions ou des interfaces poreuses. Elle porte en particulier sur le calcul des fonctions de Green, l'analyse de leurs asymptotiques et la détermination de la frontière de Martin. La théorie de la frontière de Martin permet notamment de décrire la manière dont ce type de processus s'échappe à l'infini en tenant compte de l'ensemble des trajectoires et fournit l'ensemble des fonctions excessives et harmoniques associées. Les techniques principales utilisées dans la thèse sont les équations fonctionnelles à noyau(x), la méthode du point col, divers outils d'analyse complexe sur des courbes elliptiques, ainsi que la méthode de compensation employée pour traiter les cas des processus dégénérés. Les asymptotiques des fonctions de Green pour les processus étudiés sont alors explicites, tout comme la frontière de Martin qui encode l'ensemble des fonctions harmoniques.

Mots clefs: Frontière de Martin, Equations fonctionnelles à noyau, Mouvement Brownien réfléchi obliquement, Méthode du point col, Approche par compensation, Diffusion avec barrière perméable, Transformées de Laplace.