Math & Stats: Colloquium Announcement - Renouncing the Sample Space and Revisiting the Foundations of Probability

Mathematics and Statistics Colloquium

Time and Place: March 27, 3:30 - 4:20, Arts 109

Speaker: Professor Emeritus Mikelis Bickis

Title: Renouncing the Sample Space and Revisiting the Foundations of Probability

Abstract:

Kolmogorov laid the foundations of probability in measure theory, defining events as measurable subsets of a sample space. Such a setup may be easy to visualize for games of chance but is problematic for real-life situations where it may be difficult to articulate exactly what are the elements of the sample space.

An alternative theory, going back to Carath'eodory~(1963) and advocated by his student Kappos~(1971), considers events as primitive, forming a Boolean algebra independent of any underlying sample space. Simple random variables can be defined as functions on a partition. A natural order then defines a vector lattice whose completion gives rise to continuous random variables, which are no longer viewed as functions on a sample space but as order limits of simple random variables.

Even more radically, one can consider random variables themselves as primitive: Undefined objects interpreted as unknown numbers admitting arithmetic operations, but which are only partially ordered. Events can then be identified as idempotent random variables. Probability can be introduced by postulating a convex cone of random variables that are expected to be positive. Williams~(2007), building on de Finetti's~(1974-75) foundation, proposed that this cone of random variables be given a behaviouristic interpretation as a set of gambles that you are disposed to accept because they are expected to give a positive payoff. This idea of a cone of acceptable gambles was adopted Walley~(1991), and gamble has become synonymous with random variable in subsequent literature. The expectation or emph{prevision} of a gamble is then defined as the supremum price you will pay to accept the gamble.

De Finetti viewed prevision as a linear function, meaning that the cone of acceptable gambles is in fact a half-space. Williams and Walley allowed for indecision by not requiring that either a gamble or its negative be acceptable. The consequent emph{lower prevision}, thoroughly examined by Troffaes and de Cooman~(2014), now becomes a superadditive function which generalizes the classical theory of expectation.

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