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
