Conditional expectation as an information-dependent prediction

Suppose the sample space is partitioned into observable groups. An observer knows which group occurred but cannot distinguish outcomes inside it. On each group of positive probability, average X using probabilities conditional on that group. The resulting predictor is constant within the group because it cannot exploit distinctions the observer lacks. Unequal outcome probabilities require weighted averages. Conditioning does not mean deleting uncertainty by replacing the unknown outcome with the most likely member of its group.

A sigma algebra encodes this information more generally: it lists the events whose occurrence the observer can determine. A random variable measurable with respect to that sigma algebra can be calculated from the information. If X itself is observable, its conditional expectation is X almost surely. If X is independent of the information and integrable, its conditional expectation is its ordinary mean almost surely. These two limiting cases provide useful checks on a proposed formula before any calculation begins.

For an integrable X, a conditional expectation Y given G must be G-measurable and reproduce X's integral on every event in G. Existence follows from the Radon–Nikodym theorem; the definition does not ask for division by the probability of an individual continuous observation. It characterizes an equivalence class of predictors. Two versions can disagree on an event of probability zero while representing the same conditional expectation, so a claim of uniqueness must include the words almost surely.

This qualification matters when someone asks for a conditional value at an exceptional observation. The general definition alone may leave that value unspecified. A chosen regular conditional distribution, a continuous version, or other structure can supply a meaningful convention, but that convention is additional information. An almost-sure identity is sufficient for expectations and probability statements. It is not permission to assign a unique numerical prediction at every theoretically possible input without checking the version being used.

If H contains less information than G, first averaging given G and then averaging given H gives the same result as averaging X directly given H. This tower property uses nested sigma algebras. It does not say that conditioning in arbitrary orders commutes. Also, a bounded G-measurable factor can be pulled through the conditional expectation because it is already known to the observer. Products of two unknown random variables cannot generally be split into products of their conditional means.

When X has finite second moment, conditional expectation is the orthogonal projection onto the G-measurable square-integrable random variables. Its residual has zero inner product with every such predictor, so the Pythagorean identity proves minimum mean-square error. This geometric interpretation requires L2, whereas the defining integral characterization only requires L1. The distinction is useful in heavy-tailed models: an average prediction may exist even when a squared-error objective is infinite and the projection argument cannot be used. Conditioning also preserves nonnegativity: if X is nonnegative almost surely, a valid conditional predictor cannot be negative on a positive-probability observable event.