Conditional expectation

01Average within an observed case#

A conditional mean is an expectation under a conditional distribution. Once a case has been observed, use the probabilities within that case to weight the remaining possible values.

Write $p_y(x)=P(X=x\mid Y=y)$ for the conditional probabilities. Weight each possible value by its conditional probability:

Different observed values of $Y$ can produce different means. Before observing $Y$, the expression $E[X\mid Y]$ is a rule that maps each case to its conditional mean.

02Use the conditional probabilities#

Suppose $X$ takes values $0,4$. Given $Y=a$, their conditional probabilities are $0.75,0.25$. The conditional mean is $0(0.75)+4(0.25)=1$.

Given $Y=b$, suppose the probabilities are $0.25,0.75$. The conditional mean is now $3$. The outcomes are the same; the conditional weights changed.

03A group average is still an average#

A conditional mean of $1$ does not say the next $X$ will equal $1$: only zero and four can occur in this example. It is the weighted average under the stated observation.

A regression model may predict an outcome's conditional mean from input features. This lesson assumes the conditional distribution is supplied; learning that distribution or fitting a prediction function is a separate task.

If a joint table is supplied instead, first obtain the conditional probabilities using the earlier normalization rule. Joint masses are not ready-made conditional weights. The conditioning case must have positive probability for that finite-table calculation.