SIPTA seminar- M. Troffaes: Fundamentally finitary foundations for probability & bounded probability

ABSTRACT: Bounded probability uses sets of probability measures, specified through bounds on expectations, to represent states of severe uncertainty. This approach has been successfully applied to a wide range of fields where risk under severe uncertainty is a concern.

Following Walley’s work from the 90’s, the canonical interpretation of these bounds has been through betting. However, in high-risk situations where the assessor themselves is at risk, various authors have argued that the betting interpretation of probability, and therefore also of lower previsions, cannot be applied. For this reason, in 2006, Lindley suggested an alternative interpretation of probability, based on urns, which mathematically leads to a theory of rational valued probabilities for finite possibility spaces.

Independently, in the 80’s, Nelson proposed a radically elementary probability theory based on real-valued probability mass functions for arbitrary possibility spaces, demonstrating this theory could recover, and simplify the formulation of, many important probabilistic results. This includes the functional central limit theorem that characterizes Brownian motion, and thereby Nelson removes all of the usual technicalities that arise in the measure theoretic approach to Brownian motion. Indeed, his approach needs no measure theory, a feature shared by the theories from De Finetti and Walley. Interestingly, Nelson’s approach also incorporates a fundamental notion of ambiguity, and embraces finite additivity, though these aspects are not often pointed out. Unfortunately, despite its beauty and elegance, Nelson’s programme failed to gain large traction to this day, perhaps partly because it relies on extending ZFC in a way that may first seem bizarre and counterintuitive.

In this talk, I revisit Lindley’s interpretation in the context of probability bounding. In doing so, I provide an alternative interpretation of lower previsions, which leads to new expressions for consistency (called avoiding sure loss) and inference (called natural extension). Unlike Walley’s approach, the duality theory that follows from this interpretation does not need the ultrafilter principle, and is purely constructive. A key corollary from these results is that every conditional probability measure (even finitely additive ones) can be represented by a net of probability mass functions, establishing that Nelson’s programme is universal: there is no probability measure that cannot be modelled by his approach. I reflect on what this means for practical probabilistic modelling and inference, and whether perhaps, in Nelson’s spirit, it is worthwhile to replace probability measures with probability mass functions as a foundation for probability and bounded probability, and to treat sigma-additivity as a sometimes welcome but often unnecessary by-product of idealization.

This talk is part of a series of seminars on imprecise probabilities that are organized by SIPTA, the "Society for Imprecise Probabilities: Theories and Applications". We also organize conferences and schools, provide documentation and maintain a mailing list and blog. More information is available at http://sipta.org. Info on the SIPTA seminars in particular is available at http://sipta.org/events/sipta-seminars

Contents
00:00 - Start
03:32 - Introduction
05:57 - Recap on de Finetti and Williams
23:47 - Lindley's interpretation of probability
33:29 - Nelson's radically elementary probability theory
43:38 - Urn interpretation of bounded probability
55:48 - Conclusions