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Durham University

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Publication details for Matthias Troffaes

Troffaes, Matthias C. M. & Goldstein, Michael (2021). 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

Author(s) from Durham

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.