Notes on conditional previsions

The personalist conception of probability is often explicated in terms of betting rates acceptable to an individual. A common approach, that of de Finetti for example, assumes that the individual is willing to take either side of the bet, so that the bet is “fair” from the individual’s point of view. This can sometimes be unrealistic, and leads to difficulties in the case of conditional probabilities or previsions. An alternative conception is presented in which it is only assumed that the collection of acceptable bets forms a convex cone, rather than a linear space. This leads to the more general conception of an upper conditional prevision. The main concerns of the paper are with the extension of upper conditional previsions. The main result is that any upper conditional prevision is the upper envelope of a family of additive conditional previsions.