On general conditional random quantities

Giuseppe Sanfilippo, Veronica Biazzo, Angelo Gilio, Giuseppe Sanfilippo

Risultato della ricerca: Paper

2 Citazioni (Scopus)

Abstract

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.
Lingua originaleEnglish
Stato di pubblicazionePublished - 2009

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Linearity
Conditioning
Theorem

All Science Journal Classification (ASJC) codes

  • Statistics and Probability

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Sanfilippo, G., Biazzo, V., Gilio, A., & Sanfilippo, G. (2009). On general conditional random quantities.

On general conditional random quantities. / Sanfilippo, Giuseppe; Biazzo, Veronica; Gilio, Angelo; Sanfilippo, Giuseppe.

2009.

Risultato della ricerca: Paper

Sanfilippo, G, Biazzo, V, Gilio, A & Sanfilippo, G 2009, 'On general conditional random quantities'.
Sanfilippo G, Biazzo V, Gilio A, Sanfilippo G. On general conditional random quantities. 2009.
Sanfilippo, Giuseppe ; Biazzo, Veronica ; Gilio, Angelo ; Sanfilippo, Giuseppe. / On general conditional random quantities.
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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.

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