Let's see: Conditional expectation

Thursday, January 5, 2023

Let $X$ be a sample space, $F$ be a sigma algebra and $G$ a sub-sigma algebra of $F$ .

Conditional expectation is a projection of random variables,

E : L^{2} (X, F) \to L^{2} (X, G)

. The operator can be extended to

L^{1} (X, F)

and is a Markov operator.

How is it related to conditional probability?

Let

A

be an event,

{B_{i}}

be a decomposition of

X

and

G

be the sub-sigma algebra generated by

{B_{i}}

. The conditional expectation of the indicator function

1_{A}

is a linear combination of

1_{B_{i}}

. Here the coefficients are the conditional probabilities

P (A | B_{i})

! In this sense, the conditional expectation packages all the conditional probabilities together.

More generally, given a

C^{*}

-subalgebra

B

of a

C^{*}

-algebra

A

having the same unit, conditional expectation is a positive, surjective, unital operator

T : A \to B

such that

T^{2} = T

Rotation algebra is the universal

C^{*}

-algebra generated by unitaries

u

and

v

with

u v = e^{2 π i θ} v u

. Cuntz algebra is the universal

C^{*}

-algebra generated by isometries

S_{i}

where summation of

\sum_{i} S_{i} S_{i}^{*} = 1

Conditional expectation is used to prove the simplicity of rotation algebra and Cuntz algebra.

References:

* Functional Analysis for Probability and Stochastic Processes, Adam Bobrowski

Let's see