Durham University, Department of Mathematical Sciences
Durham, United Kingdom
Tuesday 14 to Saturday 18 July 2009


Veronica Biazzo, Angelo Gilio, Giuseppe Sanfilippo

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 $\pr{(XH|K)} =\pr{(X|HK)P(H|K)}$, by simply using linearity of prevision. Then, we consider the notion of general conditional prevision $\pr(X|Y)$, 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 $\pr(X|Y)$ when $Y$ has no negative values and when $Y$ has no positive values. Finally, we give some results on coherence of $\pr(X|Y)$ when $Y$ assumes both positive and negative values. In order to illustrate critical aspects and remarks we examine several examples.

Keywords. conditional events, general conditional random quantities, general conditional prevision assessments, generalized compound prevision theorem, iterated conditioning, strong generalized compound prevision theorem.

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Authors addresses:

Veronica Biazzo
Dipartimento di Matematica e Informatica
Universita degli Studi di Catania
Citta Universitaria
Viale A. Doria 6
95152 Catania

Angelo Gilio
Dipartimento di Metodi e Modelli Matematici
Via A. Scarpa 16, 00161 Roma (Italy)

Giuseppe Sanfilippo
Dipartimento di Scienze Statistiche e Matematiche ''S.Vianelli''
Universita' degli Studi di Palermo
Viale delle Scienze ed.13
90128 Palermo,Italy

E-mail addresses:

Veronica Biazzo
Angelo Gilio
Giuseppe Sanfilippo

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