regular conditional probability

Introduction

Suppose $(\Omega,\mathcal{F},P)$ is a probability space and $B\in\mathcal{F}$ be an event with $P(B)>0$ . It is easy to see that $P_{B}:\mathcal{F}\to[0,1]$ defined by

P_{B}(A):=P(A|B),

the conditional probability of event $A$ given $B$ , is a probability measure defined on $\mathcal{F}$ , since:

1.

$P_{B}$ is clearly non-negative;
2.

$P_{B}(\Omega)=\displaystyle{\frac{P(\Omega\cap B)}{P(B)}=\frac{P(B)}{P(B)}=1}$ ;
3.

$P_{B}$ is countably additive: for if $\{A_{1},A_{2},\ldots\}$ is a countable collection of pairwise disjoint events in $\mathcal{F}$ , then

$P_{B}(\bigcup_{i=1}^{\infty}A_{i})=\frac{P\big{(}B\cap(\bigcup A_{i})\big{)}}{% P(B)}=\frac{P\big{(}\bigcup(B\cap A_{i})\big{)}}{P(B)}=\frac{\sum P(B\cap A_{i% })}{P(B)}=\sum_{i=1}^{\infty}P_{B}(A_{i}),$

as $\{B\cap A_{1},B\cap A_{2},\ldots\}$ is a collection of pairwise disjoint events also.

Regular Conditional Probability

Can we extend the definition above to $P_{\mathcal{G}}$ , where $\mathcal{G}$ is a sub sigma algebra of $\mathcal{F}$ instead of an event? First, we need to be careful what we mean by $P_{\mathcal{G}}$ , since, given any event $A\in\mathcal{F}$ , $P(A|\mathcal{G})$ is not a real number. And strictly speaking, it is not even a random variable, but an equivalence class of random variables (each pair differing by a null event in $\mathcal{G}$ ).

With this in mind, we start with a probability measure $P$ defined on $\mathcal{F}$ and a sub sigma algebra $\mathcal{G}$ of $\mathcal{F}$ . A function $P_{\mathcal{G}}:\mathcal{G}\times\Omega\to[0,1]$ is a called a regular conditional probability if it has the following properties:

1.
for each event $A\in\mathcal{G}$ , $P_{\mathcal{G}}(A,\cdot):\Omega\to[0,1]$ is a conditional probability (http://planetmath.org/ProbabilityConditioningOnASigmaAlgebra) (as a random variable) of $A$ given $\mathcal{G}$ ; that is,
1. (a)
  
  $P_{\mathcal{G}}(A,\cdot)$ is $\mathcal{G}$ -measurable (http://planetmath.org/MathcalFMeasurableFunction) and
2. (b)
  
  for every $B\in\mathcal{G}$ , we have $\displaystyle\int_{B}P_{\mathcal{G}}(A,\cdot)dP=P(A\cap B).$
2.

for every outcome $\omega\in\Omega$ , $P_{\mathcal{G}}(\cdot,\omega):\mathcal{G}\to[0,1]$ is a probability measure.

There are probability spaces where no regular conditional probabilities can be defined. However, when a regular conditional probability function does exist on a space $\Omega$ , then by condition 2 of the definition, we can define a “conditional” probability measure on $\Omega$ for each outcome in the sense of the first two paragraphs.

Title	regular conditional probability
Canonical name	RegularConditionalProbability
Date of creation	2013-03-22 16:25:24
Last modified on	2013-03-22 16:25:24
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	8
Author	CWoo (3771)
Entry type	Definition
Classification	msc 60A99