Question

Let (Ω,F,P) be a probability space and G⊆F a sub- σ-algebra. Assume furthermore that E[X2]<[infinity] (i) Show that E[(X−Y)2]=E[(X−E[X∣G])2]+E[(E[X∣G]−Y)2] holds for all Y:(Ω,G)→(R,B(R)) with E[Y2]<[infinity] (i.e. all square-integrable, G B(R)-measurable (!) random variables Y ). (ii) Conclude that the expected square distance to X is minimized among the class of squareintegrable, G−B(R)-measurable random variables Y by the choice Y=E[X∣G]. What does this result mean?

Answer 1

For (i),
`\(E[(X−Y)^2]=E[(X−E[X∣G])^2]+E[(E[X∣G]−Y)^2]\)`holds for all square-integrable,
`\(G B(R)\)-`measurable random variables
`\(Y\).` For (ii), the choice
`\(G B(R)\)-` minimizes the expected square distance to
`\(X\)` among square-integrable,
`\(G B(R)\)-`measurable random variables
`\(Y\).`

Step-by-step explanation

The equation
`\(E[(X−Y)^2]=E[(X−E[X∣G])^2]+E[(E[X∣G]−Y)^2]\)` (i) represents the law of total variance. When broken down, it elucidates that the total squared difference between
`\(X\)` and
`\(Y\)` is composed of the squared difference between
`\(X\)` and its conditional expectation given \(G\) plus the squared difference between the conditional expectation of
`\(X\)`given
`\(G\)`and
`\(Y\)`. This showcases how the variability of
`\(X\)`can be decomposed based on the information available in
`\(G\).`

The choice
`\(Y=E[X∣G]\)` in (ii) is significant as it results in the minimum expected square distance to
`\(X\)` among
`\(G−B(R)\)-`measurable random variables
`\(Y\).` By setting
`\(Y\)` as the conditional expectation of
`\(X\)` given
`\(G\),`we minimize the variability captured in
`\(Y\)` with respect to
`\(X\)`within the class of measurable random variables, highlighting that
`\(E[X∣G]\)` is the best predictor of
`\(X\)` within the information captured by
`\(G\).`

This result implies that, given the available information in the best approximation or prediction for
`\(X\) is \(E[X∣G]\).`It reflects the efficiency of the conditional expectation in minimizing the expected square distance between a random variable and its approximation, showing its significance in statistical estimation within the given probability space and sub-σ-algebra.