UE APM_51056_EP | Catalogue 2025-2026

Descriptif

Course description. This course provides an advanced course on probability theory and stochastic processes essential for modeling a variety of real-world scenarios and at the basis of Machine Learning theory. Students will become experts in the language of probability theory, enabling them to effectively analyze and address complex challenges in both pure and applied sciences. In particular, this course will focus on the problem of generative models and Monte Carlo methods. Expectations for student learning outcomes. The course has three objectives. The first is to present the foundations of probabilities based on the theory of abstract measure; on this occasion, we construct probability spaces, probabilities on measured spaces, inte- gration on abstract spaces, and we provide the essential properties of the integral. The second is to present and provide analysis of Monte Carlo methods and their Markov Chain version. Finally, the course will be concluded by a brief introduction to generative models. On successful completion of this course, a student will be able to: • Apply the fundamental concepts of probability theory and explain its position in modern statistics, machine learning, and applied contexts. • Apply basic Monte Carlo methods. • Solve basic problems in machine learning and computational statistics relating to probability theory. • Solve complex problems involving stochastic processes. Pre-requisites: bachelor-level knowledge in statistics, probability, linear algebra and calculus. Assessment: Exam and Lab Plan for the Course: Lectures 1–2: Basics of integration and measure theory, application to statistics Lectures 3–4: Monte Carlo methods Lectures 5–6: Conditional distributions and stochastic process Lectures 7–8: Markov chains and MCMC Lecture 9: Introduction to Generative models

36 heures en présentiel

Diplôme(s) concerné(s)

M1 MJH - Mathématiques Jacques Hadamard

Format des notes

Numérique sur 20

Littérale/grade réduit

Méthodes pédagogiques

\- 9 cours magistraux de 2h - 9 séances d'exercice de 2h, dont certaines sur ordinateur portable pour mettre en oeuvre les algorithmes de simulation. Les codes seront écrits en Python mais aucun preréquis Python n'est demandé: une mini-formation Python, un tutoriel et des démonstrations de simulation seront fournis au début du cours. Des exemples d'applications industrielles seront donnés pendant les cours.