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Foundations for temporal reasoning using lower previsions without a possibility space

Troffaes, Matthias C. M. and Goldstein, Michael (2022) 'Foundations for temporal reasoning using lower previsions without a possibility space.', Theory and Decision Library A: Rational Choice in Practical Philosophy and Philosophy of Science .


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.

Item Type:Article
Full text:Publisher-imposed embargo
(AM) Accepted Manuscript
File format - PDF
Publisher Web site:
Date accepted:05 January 2021
Date deposited:26 February 2021
Date of first online publication:No date available
Date first made open access:No date available

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