Description du sujet de thèse

Classical inference tools, in particular hypothesis tests and confidence intervals, can dramatically fail when applied to data-driven statistical models. Post-selection inference refers to a set of recent research works that design and analyze statistical methods tailored to these data-driven models. In particular, [3] addresses Gaussian linear models and [2] provides extensions to non-linear non-Gaussian settings, based on asymptotic arguments.
The goal of the PhD project is to extend post-selection inference to latent variables models. These models have become the method of choice in a wide range of applications [4, 6, 8] and are the object of many recent contributions [1, 5]. Nevertheless, post-selection inference guarantees are currently missing for them, while model selection often takes place in practice [7, 9].
This extension, relying on [2], will necessitate to obtain uniform joint central limit theorems for parameter estimators with latent variables. Also, from a computational point of view, parameter estimation will be performed thanks to the Expectation Maximization (EM) algorithms and their extensions. This will also necessitate mathematical developments to account for the post-selection inference context.

[1] P. Abry, J. Chevallier, G. Fort, and B. Pascal. Pandemic intensity estimation from stochastic approximation-based algorithms. In 2023 IEEE 9th International Workshop on Computational Ad- vances in Multi-Sensor Adaptive Processing (CAMSAP), pages 356–360. IEEE, 2023.
[2] F. Bachoc, D. Preinerstorfer, and L. Steinberger. Uniformly valid confidence intervals post-model- selection. The Annals of Statistics, 48(1):440–463, 2020.
[3] R. Berk, L. Brown, A. Buja, K. Zhang, and L. Zhao. Valid post-selection inference. The Annals of Statistics, pages 802–837, 2013.
[4] D. M. Blei. Build, compute, critique, repeat: Data analysis with latent variable models. Annual Review of Statistics and Its Application, 1(1):203–232, 2014.
[5] J. Chevallier, V. Debavelaere, and S. Allassonniere. A coherent framework for learning spatiotemporal piecewise-geodesic trajectories from longitudinal manifold-valued data. SIAM Journal on Imaging Sciences, 14(1):349–388, 2021.
[6] B. Everett. An introduction to latent variable models. Springer Science & Business Media, 2013.
[7] S. Lotfi, P. Izmailov, G. Benton, M. Goldblum, and A. G. Wilson. Bayesian model selection, the marginal likelihood, and generalization. In International Conference on Machine Learning, pages 14223–14247. PMLR, 2022.
[8] B. O. Muth ́en. Beyond SEM: General latent variable modeling. Behaviormetrika, 29(1):81–117, 2002.
[9] Y.-Q. Zhang, G.-L. Tian, and N.-S. Tang. Latent variable selection in structural equation models.
Journal of Multivariate Analysis, 152:190–205, 2016.

Contexte de travail

The PhD student will be located at the Institut de Mathématiques de Toulouse (IMT). The thesis will be supervised jointly by François Bachoc and Juliette Chevallier (Institut de Mathématiques de Toulouse). The PhD project will be funded by the QHTHY project involving industrial actors. The selected PhD student will have the option (non-mandatory) to attend workshops with these industrial actors and to address real data sets from the QUTHY project. The thesis will last three years, starting on October 1, 2025.

Contraintes et risques

The thesis will be attached to the Institut de Mathématiques de Toulouse. Several short trips to Marseilles and Cadarache are possible, in the frame of the QUTHY project.

Informations complémentaires

We are seeking for candidates with a degree in mathematics, with a specialization in probability, statistics, machine learning or applied mathematics. Solid theoretical skills are expected.