Treffer: Multilevel methods for Bayesian learning and applications to biological datas ; Méthodes multilevel pour l’apprentissage bayésien et applications aux données biologiques

This thesis is the result of a partnership between the Mathematics laboratory of Angers (LAREMA) and the SIRIC-ILIAD (Site of Integrated Research on Cancer) Nantes-Angers. The objective is to develop efficient numerical methods for Bayesian learning of cancer data. In addition to the statistical difficulty related to the small number of individuals compared to the number of data acquired per patient, the large dimension strongly impacts the efficiency of the numerical methods. Then, it requires the development of new methods able to apprehend the dimension. After an introduction presenting the existing tools and results,the first work of this thesis introduces a new multilevel algorithm. First described in a general framework, the complexity of this algorithm is computed more precisely for Langevin diffusions satisfying uniform convexity assumptions. In addition to the explicit description of the dependencies in the dimension, these results improve those existing in the literature. In the second step, we try to weaken the uniform convexity assumption in order to meet some statistical objectives. In this difficult framework, two techniques are studied. In the first one, the idea is to add a strongly convex component to the weakly convex potential in order to use the results of the first part. In the second part, we study an intermediate framework between uniform and weak convexity. After showing results concerning the exponential moments or the distance in a long time to the diffusion, we show that the multilevel algorithm fits in this framework. It allows us to approach the estimator with an explicit complex-ity in terms of the parameters. The last work consists of the application of the previously mentioned methods to real data. Thus, we study a dataset containing genomic data of breast cancer patients. After a dimension reduction, we compute the posterior mean associated with alogistic regression to predict the response to treatments. ; Cette thèse est issue d’un partenariat entre le laboratoire de ...