In the Wikipedia article for conditional expectation, conditional probability is defined in terms of conditional expectation.
-
Given a sub sigma algebra of the one
on a probability space.Given a probability space $(\Omega,
\mathcal{F}, P)$, a conditional
probability $P(A \mid \mathcal{B})$
of a measurable subset $A \in
\mathcal{F}$ given a sub sigma
algebra $\mathcal{B}$ of
$\mathcal{F}$, is defined as the
conditional expectation $E(A \mid
\mathcal{B})$ of indicator function
$1_A$ of $A$ given $\mathcal{B}$,
i.e. $$ P(A \mid \mathcal{B}): = E(1_A
\mid \mathcal{B}), \forall A \in
\mathcal{F}.$$So actually the conditional
probability $P(\cdot \mid
\mathcal{B})$ is a mapping $: \Omega
\times \mathcal{F} \rightarrow
\mathbb{R}$.A conditional probability $P(\cdot
\mid \mathcal{B})$ is called regular if
$P(\cdot|\mathcal{B})(\omega),
\forall \omega \in \Omega$ is also a
probability measure.Question:
What are some
necessary and/or sufficient
conditions for a conditional
probability $P(\cdot \mid
\mathcal{B})$ to be regular? -
Given a random variable on a probability space.
Suppose $(\Omega, \mathcal{F}, P)$
is a probability space and $(U,
\mathcal{\Sigma})$ is a measurable
space. There seem to be two ways of
defining the conditional expectation
$E(X\mid Y)$ of a r.v. $X: \Omega
\rightarrow \mathbb{R}$ given
another r.v. $Y: \Omega \rightarrow
U$, either as a
$\sigma(Y)$-measurable mapping $:
\Omega \rightarrow \mathbb{R}$, or
as a $\Sigma$-measurable mapping $:
U \rightarrow \mathbb{R}$, as in
my previous post.If one let $X$ to be the indicator
function $1_A$ for some $A \in
\mathcal{F}$, one can similarly
define $E(1_A \mid Y)$ to be
conditional probability of $A$ given
$Y$, denoted as $P(A\mid Y)$.
Therefore $P(\cdot \mid Y)$ is a
mapping $: \Omega \times \mathcal{F}
\rightarrow \mathbb{R}$ or a mapping
$: U \times \mathcal{F} \rightarrow
\mathbb{R}$.Questions:
(1). What are some necessary and/or
sufficient conditions for $P(\cdot
\mid Y)$ to be regular, i.e. to be a
mapping $: \Omega \rightarrow \{
\text{probability measures on
}(\Omega, \mathcal{F}) \}$ or a
mapping $: U \rightarrow \{
\text{probability measures on
}(\Omega, \mathcal{F}) \}$?(2). Under what kinds of conditions, will
$P(X \mid Y)$ defined as above be
equal to the ratio $\frac{P(X, Y)}{P(Y)}$, the
definition used in elementary
probability?
Thanks and regards! References (links or books) will also be appreciated!
Best Answer
Conditional probabilities do not give a unique function on the sample space. Since conditional expectations are only defined up to a measure zero set and one has to make an uncountable number of selections, the essential problem is whether one can "glue" them together in coherent way, so that you can actually calculate conditional probabilities by integrating the function. There are several notions of regular conditional probabilities and this paper by Faden gives necessary and sufficient conditions for some of them. For the particular version you mentioned, little is known about necessary conditions. The strongest results on the existence of regular conditional probabilities can be found in this paper by Pachl, but he only requires them to be measurable with respect to the completion of the measure. The machinery he uses is rather sophisticated, his method is based on using a lifting that he then shows (under some condition, compactness) to give a countably additive probability. The most extensive resource on conditional probabilities is probably the book Conditional Measures and Applications by M.M. Rao. The book is not recommended for its readability. Your question is addressed in chapter 3 in a comprehensive manner.