Imprecise Probabilistic Machine Learning: Being Precise About Imprecision

This talk is divided into two parts. I will first introduce the field of “Imprecise Probabilistic Machine Learning”, from its inception to modern-day research and open problems, including motivations and clarifying examples. In the second part, I will present some recent results that I've derived on Imprecise Markov Processes. I will introduce the concept of an imprecise Markov semigroup Q. It is a tool that allows to represent ambiguity around both the initial and the transition probabilities of a Markov process via a compact collection of plausible Markov semigroups, each associated with a (different, plausible) Markov process. I will use techniques from geometry, functional analysis, and (high dimensional) probability to study the ergodic behavior of Q. I will show that, if the initial distribution of the Markov processes associated with the elements of Q is known and invariant, under some conditions that also involve the geometry of the state space, eventually the ambiguity around their transition probability fades. I call this property ergodicity of the imprecise Markov semigroup, and I will relate it to the classical notion of ergodicity. I will present ergodicity when the state space is Euclidean or a Riemannian manifold. The importance of my findings for the fields of machine learning and computer vision will also be discussed.

This talk is based on the following work, https://arxiv.org/abs/2405.00081, where I also extend the results to the case where the state space is an arbitrary measurable space.