Advances in Choquet theories

Choquet theory and the theory of capacities, both initiated by French mathematician Gustave Choquet, share the heuristic notion of studying the extrema of a convex set in order to give interesting results regarding its elements. In this work, we put to use Choquet theory in the study of finite mixture models and the theory of capacities in studying severe uncertainty.

In chapter 2, we show how by combining a classical non-parametric density estimator based on a Pólya tree with techniques from Choquet theory, it is possible to retrieve the weights of a finite mixture model. We also give the rate of convergence of the Pólya tree posterior to the Dirac measure on the weights.

In chapter 3, we introduce dynamic probability kinematics (DPK), a method for an agent to mechanically update subjective beliefs in the presence of partial information. We then generalize it to dynamic imprecise probability kinematics (DIPK), which allows the agent to express their initial beliefs via a set of probabilities. We provide bounds for the lower probability associated with the updated probability sets, and we study the behavior of the latter, in particular contraction, dilation, and sure loss. Examples are provided to illustrate how the methods work. We also formulate in chapter 4 an ergodic theory for the limit of the sequence of successive DIPK updates. As a consequence, we formulate a strong law of large numbers.

Finally, in chapter 5 we propose a new, more general definition of extended probability measures ("probabilities" whose codomain is the interval [-1,1]). We study their properties and provide a behavioral interpretation. We use them in an inference procedure, whose environment is canonically represented by a probability space, when both the probability measure and the composition of the state space are unknown. We develop an ex ante analysis - taking place before the statistical analysis requiring knowledge of the state space - in which we progressively learn its true composition. We describe how to update extended probabilities in this setting, and introduce the concept of lower extended probabilities. We provide two examples in the fields of ecology and opinion dynamics.