TY - CONF

T1 - On general conditional random quantities

AU - Sanfilippo, Giuseppe

AU - Gilio, Angelo

AU - Sanfilippo, Giuseppe

AU - Biazzo, Veronica

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Conditional events

KW - general conditional prevision assessments

KW - general conditional random quantities

KW - generalized compound prevision theorem

KW - iterated conditioning

KW - strong generalized compound prevision theorem

KW - Conditional events

KW - general conditional prevision assessments

KW - general conditional random quantities

KW - generalized compound prevision theorem

KW - iterated conditioning

KW - strong generalized compound prevision theorem

UR - http://hdl.handle.net/10447/47606

UR - http://www.sipta.org/isipta09/proceedings/031.html

M3 - Other

SP - 51

EP - 60

ER -