On general conditional random quantities

In the first part of this paper, recalling a general discussion on iterated conditioning given by de Finetti in the appendix of his book, vol. 2, we give a representation of a conditional random quantity X|HK as (X|H)|K. In this way, we obtain the classical formula P(XH|K) = P(X|HK)P (H|K), by simply using linearity of prevision. Then, we consider the notion of general conditional prevision P(X|Y ), where X and Y are two random quantities, introduced in 1990 in a paper by Lad and Dickey. After recalling the case where Y is an event, we consider the case of discrete finite random quantities and we make some critical comments and examples. We give a notion of coherence for such more general conditional prevision assessments; then, we obtain a strong generalized compound prevision theorem. We study the coherence of a general conditional prevision assessment P(X|Y ) when Y has no negative values and when Y has no positive values. Finally, we give some results on coherence of P(X|Y ) when Y assumes both positive and negative values. In order to illustrate critical aspects and remarks we examine several examples.