Bandit Algorithms

2.6 Conditional expectation 35

andE[X|H] is uniquely defined only up to events ofP-measure zero, none of

this should be of a significant concern. First, we will rarely need closed form

expressions for conditional expectations, but we rather need how they relate to

other expectations, conditional or not. This is also the reason why it should not be

concerning that they are only determined up to zero probability events: Usually,

conditional expectations appear in other expectations or in statements that are

concerned with how probable some event is, making the difference between the

different ‘versions’ of conditional expectations disappear.

We close the section by summarizing some additional important properties of

conditional expectations. These follow from the definition directly and the reader

is invited to prove them in Exercise 2.19.

Theorem2.12. Let(Ω,F,P)be a probability space,G,G 1 ,G 2 ⊂ F be sub-σ-
algebras ofFandX,Y integrable random variables on(Ω,F,P). The fol lowing
hold true:

1 IfX≥ 0 , thenE[X|G]≥ 0 almost surely.

2 E[1|G] = 1almost surely.

3 E[X+Y|G] =E[X|G] +E[Y|G]almost surely.

4 E[XY|G] =YE[X|G]almost surely ifE[XY]exists andY isG-measurable.

5 ifG 1 ⊂G 2 , thenE[X|G 1 ] =E[E[X|G 2 ]|G 1 ]almost surely.

6 ifσ(X)is independent ofG 2 givenG 1 thenE[X|σ(G 1 ∪G 2 )]=E[X|G 1 ]

almost surely.

7 IfG={∅,Ω}is the trivialσ-algebra, thenE[X|G] =E[X]almost surely.

Properties 1 and 2 are self-explanatory. Property 3 generalizes the linearity of

expectation. Property 4 shows that a measurable quantity can be pulled outside

of a conditional expectation and corresponds to the property that for constants

c,E[cX]=cE[X]. Property 5 is called thetower ruleor thelaw of total

expectations. It says that the fineness ofE[X|G 2 ] is obliterated when taking the

conditional expectation with respect toG 1. Property 6 relates independence and

conditional expectations and it says that conditioning on independent quantities

does not give further information on expectations. Here, the two event systemsA

andBare said to beconditionally independentof each other given aσ-algebra

Fif for allA∈ AandB∈ B,P(A∩B|F)=P(A|F)P(B|F)holds almost

surely. We also often say thatAis conditionally independent ofBgivenF, but

of course, this relation is symmetric. This property is often applied with random

variables:Xis said to be conditionally independent ofY givenZ, ifσ(X) is

conditionally independent ofσ(Y) givenσ(Z). In this case,E[X|Y,Z]=E[X|Z]

holds almost surely. Property 7 states that conditioning on no information gives

the same expectation as not conditioning at all.