Conditional Expectation

Conditional expectation is the best prediction of a random variable given the information in a sub-sigma-algebra, defined abstractly through the Radon-Nikodym theorem and behaving like an averaging projection that respects the available information.

Definition

The conditional expectation of an integrable random variable given a sub-sigma-algebra is the unique, up to almost-sure equality, integrable function that is measurable with respect to that sub-sigma-algebra and has the same integral as the original variable over every set in it.

Scope

The topic covers the definition of conditional expectation given a sub-sigma-algebra, its existence and almost-sure uniqueness via the Radon-Nikodym theorem, the tower, taking-out-what-is-known, and conditional-Jensen properties, the interpretation as an orthogonal projection in the space of square-integrable variables, conditional probability and regular conditional distributions, and the role of conditioning as the engine of martingales and Bayesian updating.

Core questions

• How can expectation be conditioned on information that may include events of probability zero?
• Why is conditional expectation unique only up to an almost-sure null set?
• In what sense is conditional expectation the best predictor of a random variable?
• How do the tower and pull-out properties make conditional expectation tractable?

Key concepts

• conditioning sigma-algebra
• Radon-Nikodym derivative
• tower property
• least-squares projection
• regular conditional distribution

Key theories

Existence via Radon-Nikodym

Conditional expectation exists because the measure obtained by integrating the random variable over sets of the sub-sigma-algebra is absolutely continuous with respect to the restricted probability measure, and its Radon-Nikodym derivative is the conditional expectation.

Tower property

Conditioning on a coarser sigma-algebra after conditioning on a finer one returns the coarser conditional expectation, so iterated conditioning collapses to the coarsest level; this smoothing identity is fundamental to martingale theory and filtering.

Projection characterization

For square-integrable variables, conditional expectation is the orthogonal projection onto the subspace of variables measurable with respect to the conditioning sigma-algebra, which makes it the least-squares-optimal predictor given the available information.

Clinical relevance

Conditional expectation is the formal basis of prediction and updating under uncertainty: it defines martingales, underlies the Kalman filter and nonlinear filtering, expresses Bayesian posterior means, and gives the no-arbitrage price of a contingent claim as a conditional expectation under a risk-neutral measure.

History

Kolmogorov introduced the general definition of conditional expectation with respect to a sigma-algebra in 1933, resolving the paradoxes of conditioning on null events by anchoring it in the Radon-Nikodym theorem; Doob then made it the foundation of martingale theory.

Key figures

• Andrey Kolmogorov
• Joseph L. Doob
• Johann Radon
• Otton Nikodym

Seminal works

• williams1991

Frequently asked questions

Why is conditional expectation a random variable rather than a number?

Because it must encode the predicted value for every possible state of the conditioning information; as that information varies over the sample space the predicted value varies, making conditional expectation a function measurable with respect to the conditioning sigma-algebra.

How does conditioning on a sigma-algebra generalize conditioning on an event?

Conditioning on an event of positive probability is the special case where the sub-sigma-algebra is generated by that event and its complement; the general definition extends this to information that cannot be captured by any single positive-probability event.