If we are conditioning X given the trivial sigma algebra then we get the expectation of X, its proof is trivial but intuitively what does this case represent ?
$\begingroup$
$\endgroup$
2
-
$\begingroup$ The trivial sigma algebra being $\{\emptyset, \Omega\}$ ? So no additional information? $\endgroup$– HenryCommented Mar 19 at 22:19
-
$\begingroup$ @Henry, what about the discret sigma algebra ? if we condition on it then aren't we in the same case (no additional information) ? $\endgroup$– PatCommented Mar 19 at 23:10
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
5
Let $F_0=\text{{$\emptyset, \Omega$}}$ be the trivial sigma algebra for a probability space $(\Omega, F, P)$. Let $X$ be some integrable random variable defined on $(\Omega, F, P)$.
Intuition that can be helpful for conditional expectation is to think of the operation as providing the "best guess" of a random variable given some information (the trivial sigma algebra being no additional information as Henry points out in his comment). This "best guess" is taken to be the expected value given the information available.
So when we look at the conditional expectation $\mathbb{E} [X|F_0]$, as we condition on the trivial sigma algebra, we have no additional information on the value of $X$. As a result, intuitively our "best guess" for $X$ is simply its expected value. In other words: $\mathbb{E} [X|F_0]$=$\mathbb{E} [X]$.
(I am relatively new to posting answers on this forum so happy address any comments / questions for this answer.)
-
$\begingroup$ Same follow up question, if the conditional expectation is the ''best guess'' then how can we explain that the conditional expectation of X given the discrete sigma algebra is X itself, does that mean we have all information about X ? $\endgroup$– PatCommented Mar 20 at 22:50
-
$\begingroup$ What do you mean by discrete sigma algebra? If we condition on any sigma algebra $F$ where $X$ is $F$-measurable then the conditional expectation becomes $\mathbb{E}[X|F]=X$ since the sigma algebra has enough information to fully determine $X$ and so the "best guess" is simply $X$ itself. $\endgroup$ Commented Mar 20 at 23:41
-
$\begingroup$ The discrete sigma algebra is the P(E) (set of all subsets of E). $\endgroup$– PatCommented Mar 21 at 22:09
-
$\begingroup$ In general, we may not be able to use the power set of $\Omega$ to define a measure (For example, the Lebesgue measure cannot be defined for all subsets of $\mathbb{R}$). In any case, the previous comment also covers the case of the discrete sigma algebra as any random variable would be measurable on the discrete sigma algebra. $\endgroup$ Commented Mar 22 at 0:04
-
$\begingroup$ What's exactly the reason behind the intuition of ''best guess'' ? I mean what does it mean to get information from a sigma algebra ? $\endgroup$– PatCommented Mar 22 at 1:54