Foundations for temporal reasoning using lower previsions without a possibility space

Troffaes, Matthias C.M.; Goldstein, Michael

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

• 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
• Contract Date: Jan 5, 2021