Foundations for temporal reasoning using lower previsions without a possibility space

Troffaes, Matthias C.M.; Goldstein, Michael

doi:10.1007/978-3-031-15436-2_4

Foundations for temporal reasoning using lower previsions without a possibility space

Troffaes, Matthias C.M.; Goldstein, Michael

Authors

Professor Matthias Troffaes matthias.troffaes@durham.ac.uk
Professor

Professor Michael Goldstein michael.goldstein@durham.ac.uk
Professor

Contributors

Thomas Augustin
Editor

Fabio Gagliardi Cozman
Editor

Gregory Wheeler
Editor

Abstract

We introduce a new formal mathematical framework for probability theory, taking random quantities to be the fundamental objects of interest, without reference to a possibility space, in spirit of de Finetti’s treatment of probability, Goldstein’s Bayes linear analysis, and Williams’s treatment of lower and upper previsions. The aim of our framework is to formalize temporal reasoning, where we treat future beliefs as random quantities themselves. We do this by taking random quantities to form a linear space of expressions, which we endow with structure through a linear projection operator. We then use a version of the temporal sure preference principle as a basis for inference over time. We formulate the principle in terms of desirability, and explore its implications for lower previsions. We derive an explicit expression for the natural extension of a lower prevision under the temporal sure preference principle. We establish consistency of the temporal sure preference principle with any given collection of assessments. We also derive various bounds on the natural extension. Finally, we show how we can recover standard Bayes linear calculus from our framework.

Citation

Troffaes, M. C., & Goldstein, M. (2022). Foundations for temporal reasoning using lower previsions without a possibility space. In T. Augustin, F. Gagliardi Cozman, & G. Wheeler (Eds.), Reflections on the Foundations of Probability and Statistics: Essays in Honor of Teddy Seidenfeld (69-96). (1). Springer Verlag. https://doi.org/10.1007/978-3-031-15436-2_4

Acceptance Date	Jan 5, 2021
Online Publication Date	Aug 9, 2022
Publication Date	2022
Deposit Date	Feb 15, 2021
Publicly Available Date	Aug 10, 2024
Publisher	Springer Verlag
Pages	69-96
Series Title	Theory and Decision Library A (TDLA)
Series Number	54
Edition	1
Book Title	Reflections on the Foundations of Probability and Statistics: Essays in Honor of Teddy Seidenfeld
ISBN	9783031154355
DOI	https://doi.org/10.1007/978-3-031-15436-2_4
Public URL	https://durham-repository.worktribe.com/output/1625999