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regular conditional probability∗
CWoo†
2013-03-21 21:30:55
Introduction
Suppose (Ω, F, P ) is a probability space and B ∈ F be an event with P (B) > 0.
It is easy to see that PB : F → [0, 1] defined by
PB (A) := P (A|B),
the conditional probability of event A given B, is a probability measure defined
on F, since:
1. PB is clearly non-negative;
2. PB (Ω) =
P (B)
P (Ω ∩ B)
=
= 1;
P (B)
P (B)
3. PB is countably additive: for if {A1 , A2 , . . .} is a countable collection of
pairwise disjoint events in F, then
S
S
P
∞
∞
[
P B ∩ ( Ai )
P
(B ∩ Ai )
P (B ∩ Ai ) X
PB ( A i ) =
=
=
=
PB (Ai ),
P (B)
P (B)
P (B)
i=1
i=1
as {B ∩ A1 , B ∩ A2 , . . .} is a collection of pairwise disjoint events also.
Regular Conditional Probability
Can we extend the definition above to PG , where G is a sub sigma algebra of F
instead of an event? First, we need to be careful what we mean by PG , since,
given any event A ∈ F, P (A|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 G).
With this in mind, we start with a probability measure P defined on F and
a sub sigma algebra G of F. A function PG : G × Ω → [0, 1] is a called a regular
conditional probability if it has the following properties:
∗ hRegularConditionalProbabilityi
created: h2013-03-21i by: hCWooi version: h38574i
Privacy setting: h1i hDefinitioni h60A99i
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1. for each event A ∈ G, PG (A, ·) : Ω → [0, 1] is a conditional probability (as
a random variable) of A given G; that is,
(a) PG (A, ·) is G-measurable and
Z
(b) for every B ∈ G, we have
PG (A, ·)dP = P (A ∩ B).
B
2. for every outcome ω ∈ Ω, PG (·, ω) : G → [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 Ω, then by condition 2 of the definition, we can define a “conditional” probability measure on Ω for each outcome in the sense of the first two
paragraphs.
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