Conditional expectation

Conditional expectation is a random variable representing the expected value of another random variable given specified information. It generalizes averaging over events to conditioning on a variable or information set.

Conditional probability: The probability of an event given that another event has occurred or specified information is available. For event indicators, conditional expectation is conditional probability.
Expected value: The probability-weighted average of a random variable's possible values. Conditional expectation extends ordinary expectation by restricting the available information.
Regression analysis: A statistical method for modeling the relationship between an outcome and explanatory variables. The regression function often estimates the conditional mean of an outcome given predictors.
Unconditional expectation: The average value of a random variable without conditioning on additional information. It averages across all information states rather than retaining a conditional mean.
Sigma-algebra: A collection of sets containing the sample space and closed under complements and countable unions. A sigma-algebra formalizes exactly which information the conditional expectation may use.
Random variable: A measurable function assigning numerical values to outcomes in a probability space. The quantity being averaged conditionally is itself a random variable.
Bayesian inference: A method of updating probability distributions using observed data and Bayes' theorem. Posterior predictive means are conditional expectations given observed data.
Conditional median: A median of a random variable's distribution conditional on specified information. Unlike conditional expectation, it is a quantile-based summary and need not minimize squared error.
Tower property: The identity that iterated conditional expectations over nested information sets equal conditioning on the coarser set. It lets expectations be simplified in stages as information is progressively reduced.
Measurable function: A function whose inverse images of measurable sets are measurable. Conditional expectations must be measurable with respect to the information they represent.

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