On General Conditional Random Quantities
Giuseppe Sanfilippo, Veronica Biazzo, Angelo Gilio · 2009
In the rst 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 XjHK as (XjH)jK. In this way, we obtain the classical formula P(XHjK) =P(XjHK)P (HjK), by simply using linearity of prevision. Then, we consider the notion of general conditional prevision P(XjY ), 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 nite 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(XjY ) when Y has no negative values and when Y has no positive values. Finally, we give some results on coherence ofP(XjY ) when Y assumes both positive and negative values. In order to illustrate critical aspects and remarks we examine several examples.