Improper regular conditional distributions (rcd’s) given a
$\sigma$-field $\mathscr{A}$ have the following anomalous property. For sets $A
\in \mathscr{A}, \mathrm{Pr}(A|\mathscr{A})$ is not always equal to the
indicator of $A$. Such a property makes the conditional probability puzzling as
a representation of uncertainty. When rcd’s exist and the$\sigma$-field
$\mathscr{A}$ is countably generated, then almost surely the rcd is proper. We
give sufficient conditions for an rcd to be improper in a maximal sense, and
show that these conditions apply to the tail $\sigma$-field and the
$\sigma$-field of symmetric events.
Publié le : 2001-10-14
Classification:
Completion of $\sigma$-field,
countably generated $\sigma$-field,
nonmeasurable set,
symmetric $\sigma$-field,
tail $\sigma$-field,
60A10
@article{1015345764,
author = {Seidenfeild, Teddy and Schervish, Mark J. and Kadane, Joseph B.},
title = {Improper Regular Conditional Distributions},
journal = {Ann. Probab.},
volume = {29},
number = {1},
year = {2001},
pages = { 1612-1624},
language = {en},
url = {http://dml.mathdoc.fr/item/1015345764}
}
Seidenfeild, Teddy; Schervish, Mark J.; Kadane, Joseph B. Improper Regular Conditional Distributions. Ann. Probab., Tome 29 (2001) no. 1, pp. 1612-1624. http://gdmltest.u-ga.fr/item/1015345764/