Conditional Expectation

Overview

Conditional expectation turns partial information into a prediction. Unlike ordinary expectation, which is a number, it is generally a random variable: different observed information can produce different predictions. Its defining requirement is preservation of averages over every event that the information can distinguish.

This formulation handles discrete observations, continuous observations, and information that is not conveniently represented by one numerical measurement. In particular, it does not require dividing by the probability of an individual observed value.

Prerequisites

Work on a probability space $(\Omega,\mathcal F,P)$ with an integrable real random variable $X$, meaning $E|X|<\infty$. A sub-sigma-algebra $\mathcal G\subseteq\mathcal F$ represents available information. Measurability with respect to $\mathcal G$ means an observation can be determined using that information.

Finite partitions provide the first examples. Lebesgue integration and the Radon–Nikodym theorem explain existence in general.

Definition and conventions

A conditional expectation $Y=E[X\mid\mathcal G]$ is an integrable, $\mathcal G$-measurable random variable such that

It exists and is unique up to almost-sure equality. Choosing one representative is choosing a version; values on null sets are not determined by these identities.

If $\mathcal G$ is generated by a finite partition $(A_j)$, then on each positive-probability cell,

Choose any constant value on each null partition cell, preserving $\mathcal G$-measurability. The notation $E[X\mid Z]$ abbreviates $E[X\mid\sigma(Z)]$. A pointwise expression $E[X\mid Z=z]$ requires a suitable version and is not defined by the elementary event ratio when $P(Z=z)=0$.

Intuition

Imagine hiding the precise outcome while revealing only its partition cell. All outcomes in that cell must receive the same prediction. The integral identity chooses the cell's average, so the prediction has no systematic error on any observable event.

For $X\in L^2$, this becomes an exact least-squares statement: $E[X\mid\mathcal G]$ minimizes $E[(X-Z)^2]$ over $\mathcal G$-measurable $Z\in L^2$. The second-moment assumption belongs to this interpretation, not to the basic definition.

Examples and non-examples

Let $X$ be a fair die result, and reveal whether $X\le3$. The conditional expectation is $2$ on $\{1,2,3\}$ and $5$ on $\{4,5,6\}$. On the first cell its integral is $2(1/2)=1$, agreeing with $(1+2+3)/6$.

The constant $7/2$ has the correct overall expectation but fails the first-cell test: its integral there is $7/4$, not $1$. Matching a global average is insufficient.

If $\mathcal G$ is trivial, the answer is the constant $E[X]$. If $X$ is already $\mathcal G$-measurable, the answer is $X$ itself. Independence of $X$ and $\mathcal G$ gives the first conclusion even when $\mathcal G$ contains substantial information about other variables.

Important properties

Conditional expectation is linear and preserves positivity and total expectation. If $\mathcal H\subseteq\mathcal G$, the tower property states

Nestedness matters: unrelated information sets cannot simply be substituted. For bounded $\mathcal G$-measurable $Z$, one may pull out the known factor: $E[ZX\mid\mathcal G]=Z E[X\mid\mathcal G]$.

Proof idea: verify measurability and the defining integral identities, then invoke uniqueness. Existence instead comes from applying Radon–Nikodym to $A\mapsto\int_A X\,dP$ on $\mathcal G$. In the square-integrable case, orthogonality of the residual to observable variables proves the least-squares claim.

Connections and further reading

Successive information sets produce martingales through $E[X\mid\mathcal F_n]$. For square-integrable $X$, total variance separates remaining uncertainty from variation of the prediction. See Lawler, §6.1 for the integral characterization and Li, Definition 1.19 and Proposition 1.6 for the Chinese usage and variance decomposition.