The fundamental theorem of prevision

Abstract

We study de Finetti's "fundamental theorem of probability", reformulated in the finite case as a computable linear programming problem. The theorem is substantially extended, and shown to have important implications for the subjective theory of statistical inference. It supports an operational meaning for the partial assertion of prevision via asserted bounds. We extend the theorem to apply to quantities more general than events, to allow bounds and orderings on previsions as the input to the programming problem, and even to allow bounds on conditional previsions as the input and output. In a philosophical discussion, prevision is explicitly recognized as a completion of the notion of logical assertion, introduced by Frege. Partial prevision assertions then allow the representation of weaker forms of uncertain knowledge, spanning a range of possible numerical values for a quantity between the assertion of a precise prevision value and the mere unassertive contemplation of the quantity.