Probabilities and Statistic seminar

Dans cet exposé, on construit, pour tout automate cellulaire probabiliste uni-dimensionnel exponentiellement ergodique et possédant une propriété de taux positifs, un flot CFTP (“coupling from the past”) localement défini. Plusieurs conséquences de cette construction sont discutées. (Travail exposé dans l’article arXiv:2106.07219).

Je montre comment à partir d’une étude approfondie statistique et probabiliste d’une base de données de températures moyennes mondiales, j’ai découvert des comportements climatologiques sans doute très difficilement accessibles par les techniques usuelles utilisées en climatologie.

In a recent paper, Leonid Petrov showed that the up-down chains associated to the Chinese Restaurant Process (CRP) have a scaling limit – namely, a two-parameter family of diffusions that extend the one-parameter infinitely-many-neutral-alleles diffusions of Ethier and Kurtz. There has since been considerable interest in constructing ordered analogues of Petrov’s diffusions, and it is conjectured that an ordered analogue of the up-down chains will give rise to such an object. In this talk, I’ll discuss my resolution of this conjecture (joint with Douglas Rizzolo). Our approach is mainly inspired by Petrov’s work, and involves using quasisymmetric functions to describe the transition operators.

This presentation is a summary of my PhD work. I focus on the topic of hypothesis testing, extensively studied in statistics and theoretical computer science.

I start with presenting the classical identity testing problem, in which an independent sample set X ~ q is given and one would like to determine whether q=p for some fixed known p. This problem is very related to that of estimating a distribution from a given sample set. The study of testing is relevant, because for the same fixed sample size, it is possible to test against a distribution up to a smaller separation distance than what is possible in estimation. This will give me the opportunity to describe the minimax framework which proves the theoretical optimality of statistical methods in the worst case.

I will refine the study of minimax identity testing by adding a local differential privacy condition and the interest will be in the quantitative effect of ensuring privacy. The presentation will largely be on the topic of privacy, because it bears similarities with ensuring fairness conditions.

We will also shortly consider the neighboring problem of closeness testing, where the goal remains to determine whether p=q, but only an independent sample set Y ~ p is given instead of p directly. In this context, we will go beyond a simple worst-case analysis and develop instance optimal results instead. This will highlight the interplay between one-sample testing and two-sample testing, the latter being a harder problem.

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Differentiation is the process whereby a cell acquires a specific phenotype, by differential gene expression as a function of time. This is thought to result from the dynamical functioning of an underlying Gene Regulatory Network (GRN). The precise path from the stochastic GRN behavior to the resulting cell state is still an open question. In this presentation, we detail a methodology to reduce a mechanistic model characterizing the evolution of a cell by a system of piecewise deterministic Markov processes (PDMP), to a discrete coarse-grained model on a limited number of cell types, defined as the basins of attraction of the deterministic limit. The transitions between the basins in the weak noise limit can be determined by the unique solution of an Hamilton-Jacobi equation under a particular constraint, which corresponds to the rate function associated to a Large Deviations Principle for the PDMP. We develop a numerical method for approximating the coarse-grained model parameters, and show its accuracy for a toggle-switch network. We deduce from the reduced model an analytical approximation of the stationary distribution of the PDMP system, which appears as a Beta mixture.

Les modèles d’urnes sont utilisés dans de nombreuses applications et sont un exemple fondamental de processus stochastiques de renforcement. En partant de ces modèles, nous nous intéresserons à plusieurs familles de systèmes (finis) de processus de renforcement. Différents résultats sur le comportement collectif en temps long seront présentés. La présence/absence de synchronisation sera discutée, ainsi que les vitesses de convergence en fonction de différents régimes de paramètres. Cet exposé se fonde sur des travaux en collaboration avec I. Crimaldi, P. Dai Pra, I. Minelli et M. Mirebrahimi.

Quasi-stationary distributions can be seen as the first eigenvector associated with the generator of the stochastic differential equation at hand, on a domain with Dirichlet boundary conditions (which corresponds to absorbing boundary conditions at the level of the underlying stochastic processes). Many results on the quasi-stationary distribution hold for non degenerate stochastic dynamics, whose associated generator is elliptic. The case of degenerate dynamics is less clear. In this work, together with T. Lelièvre and J. Reygner (Ecole des Ponts, France) we generalize well-known results on the probabilistic representation of solutions to parabolic equations on bounded domains to the so-called kinetic Fokker-Planck equation on bounded domains in positions, with absorbing boundary conditions. Furthermore, a Harnack inequality, as well as a maximum principle, is provided for solutions to this kinetic Fokker-Planck equation, as well as the existence of a smooth transition density for the associated absorbed Langevin dynamics. The continuity of this transition density at the boundary is studied as well as the compactness, in various functional spaces, of the associated semigroup. This work is a cornerstone to prove the consistency of some algorithms used to simulate metastable trajectories of the Langevin dynamics, for example the Parallel Replica algorithm.

Considérons la percolation de premier passage dans le réseau renormalisé Z^d/n pour d>=2 : à chaque arête e, on associe une capacité aléatoire c(e)>=0 de telle sorte que la famille (c(e))_e soit indépendante et identiquement distribuée selon une loi G. On peut interpréter cette capacité comme un débit maximal, i.e., la quantité maximale d’eau pouvant traverser l’arête par unité de temps. Considérons un domaine borné et connecté Ω de R^d et deux ensembles disjoints du bord de Ω : un part lequel l’eau peut entrer (la source) et un part lequel l’eau peut sortir (le puits). Nous nous intéressons au flot maximal : la quantité maximale d’eau pouvant entrer dans Ω par unité de temps. Un courant est une fonction sur les arêtes qui décrit la façon dont l’eau circule dans Ω. Dans cet exposé, nous présenterons un principe de grande déviation pour les courants et nous en déduirons par un principe de contraction un principe de grande déviation pour le flot maximal dans Ω.
Travail en collaboration avec Marie Théret.

A growth-fragmentation is a stochastic process representing cells with continuously growing mass, which experience sudden splitting events. Growth-fragmentations are used to model cell division and protein polymerisation in biophysics. It is interesting to ask whether these processes converge toward an equilibrium, in which the number of cells is growing exponentially and the distribution of cell sizes approaches some fixed asymptotic profile. In this work, we study a process in which the growth and splitting of an individual cell is largely independent of its mass, with the exception that the mass is bounded above, so it cannot exceed a given constant. We give precise conditions to ensure that, almost surely, the process exhibits this equilibrium behaviour, and express the asymptotic profile in terms of an underlying Lévy process.

This is joint work with Emma Horton (Inria Bordeaux).