Conditional expectation

The expected value page averaged with respect to a complete distribution, while the marginal distribution page recovered a one-variable distribution by summing out another variable. Conditional expectation instead averages the response under one conditional distribution. Suppose Y is lexical decision time and X records whether the target word is highly predictable in context.

For low predictability, let the conditional PMF be

decision time y	p_{Y\mid X}(y\mid\mathrm{low})
400 ms	.20
500 ms	.50
600 ms	.30

For high predictability, let it be

decision time y	p_{Y\mid X}(y\mid\mathrm{high})
400 ms	.60
500 ms	.30
600 ms	.10

Each table is a complete distribution of Y within one value of X.

Averaging within low predictability

The expected decision time is

\begin{aligned} \mathbb{E}[Y\mid X=\mathrm{low}] &=400(.20)+500(.50)+600(.30)\\ &=80+250+180\\ &=510\text{ ms}. \end{aligned}

Averaging within high predictability

The corresponding expectation is

\begin{aligned} \mathbb{E}[Y\mid X=\mathrm{high}] &=400(.60)+500(.30)+600(.10)\\ &=240+150+60\\ &=450\text{ ms}. \end{aligned}

These values are conditional expectations. Each expectation uses the conditional distribution of Y at one declared value of X.

For discrete variables and a value x with p_X(x)>0, provided the absolute sum is finite,

\mathbb{E}[Y\mid X=x] \equiv\sum_y y\,p_{Y\mid X}(y\mid x).

When a conditional density f_{Y\mid X}(\,\cdot\mid x) exists and the absolute integral is finite,

\mathbb{E}[Y\mid X=x] \equiv\int y f_{Y\mid X}(y\mid x)\,\mathrm{d}y.

Treating the conditional mean as a function

Define

m(x)\equiv\mathbb{E}[Y\mid X=x].

For the two predictability conditions,

m(\mathrm{low})=510

and

m(\mathrm{high})=450.

The function maps a predictor value to the center of the response distribution under that condition. Regression models later specify how this function changes with one or more predictors.

The conditional contrast is

m(\mathrm{high})-m(\mathrm{low}) =450-510 =-60\text{ ms}.

Under the constructed model, the conditional mean for high predictability is 60 ms lower than the conditional mean for low predictability.

Recovering the marginal mean from the mixture

Suppose high predictability contexts receive probability .70 and low predictability contexts receive .30. The marginal expectation is

\begin{aligned} \mathbb{E}[Y] &=.70(450)+.30(510)\\ &=315+153\\ &=468\text{ ms}. \end{aligned}

This is an application of iterated expectation:

\mathbb{E}[Y] =\mathbb{E}\!\left[\mathbb{E}[Y\mid X]\right].

The outer expectation averages the conditional means using the distribution of X.

Checking the values in base R

Code

time <- c(400, 500, 600)
mass_low <- c(.20, .50, .30)
mass_high <- c(.60, .30, .10)

mean_low <- sum(time * mass_low)
mean_high <- sum(time * mass_high)
marginal_mean <- .30 * mean_low + .70 * mean_high

c(mean_low, mean_high, marginal_mean)

The returned values are 510, 450, and 468.

Distinguishing a mixture change from a conditional change

The marginal mean can change because the proportions of the conditions change, even when the conditional distributions remain fixed. If the share of high predictability contexts changes, the marginal mean changes even when both conditional PMFs remain fixed.

Report conditional means and the condition mixture separately. A marginal difference need not imply that the within condition response process changed.

Check your understanding

Recalculate the marginal mean if high and low predictability are equally probable.
Does changing the probability of high predictability alter either conditional mean?
What does the function m(x) return?
Explain the two averaging stages in iterated expectation.

Conditional expectation compares response centers across conditions. The next page summarizes whether paired deviations of two numerical variables tend to share a direction.

--- title: "Conditional expectation" --- The [expected value page](expected-values.qmd) averaged with respect to a complete distribution, while the [marginal distribution page](marginal-distributions.qmd) recovered a one-variable distribution by summing out another variable. Conditional expectation instead averages the response under one conditional distribution. Suppose $Y$ is lexical decision time and $X$ records whether the target word is highly predictable in context. For low predictability, let the conditional PMF be | decision time $y$ | $p_{Y\mid X}(y\mid\mathrm{low})$ | |---:|---:| | 400 ms | $.20$ | | 500 ms | $.50$ | | 600 ms | $.30$ | For high predictability, let it be | decision time $y$ | $p_{Y\mid X}(y\mid\mathrm{high})$ | |---:|---:| | 400 ms | $.60$ | | 500 ms | $.30$ | | 600 ms | $.10$ | Each table is a complete distribution of $Y$ within one value of $X$. ## Averaging within low predictability The expected decision time is $$ \begin{aligned} \mathbb{E}[Y\mid X=\mathrm{low}] &=400(.20)+500(.50)+600(.30)\\ &=80+250+180\\ &=510\text{ ms}. \end{aligned} $$ ## Averaging within high predictability The corresponding expectation is $$ \begin{aligned} \mathbb{E}[Y\mid X=\mathrm{high}] &=400(.60)+500(.30)+600(.10)\\ &=240+150+60\\ &=450\text{ ms}. \end{aligned} $$ These values are [**conditional expectations**](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/04%3A_Expected_Value/4.07%3A_Conditional_Expected_Value). Each expectation uses the conditional distribution of $Y$ at one declared value of $X$. For discrete variables and a value $x$ with $p_X(x)>0$, provided the absolute sum is finite, $$ \mathbb{E}[Y\mid X=x] \equiv\sum_y y\,p_{Y\mid X}(y\mid x). $$ When a conditional density $f_{Y\mid X}(\,\cdot\mid x)$ exists and the absolute integral is finite, $$ \mathbb{E}[Y\mid X=x] \equiv\int y f_{Y\mid X}(y\mid x)\,\mathrm{d}y. $$ ## Treating the conditional mean as a function Define $$ m(x)\equiv\mathbb{E}[Y\mid X=x]. $$ For the two predictability conditions, $$ m(\mathrm{low})=510 $$ and $$ m(\mathrm{high})=450. $$ The function maps a predictor value to the center of the response distribution under that condition. Regression models later specify how this function changes with one or more predictors. The conditional contrast is $$ m(\mathrm{high})-m(\mathrm{low}) =450-510 =-60\text{ ms}. $$ Under the constructed model, the conditional mean for high predictability is 60 ms lower than the conditional mean for low predictability. ## Recovering the marginal mean from the mixture Suppose high predictability contexts receive probability $.70$ and low predictability contexts receive $.30$. The marginal expectation is $$ \begin{aligned} \mathbb{E}[Y] &=.70(450)+.30(510)\\ &=315+153\\ &=468\text{ ms}. \end{aligned} $$ This is an application of iterated expectation: $$ \mathbb{E}[Y] =\mathbb{E}\!\left[\mathbb{E}[Y\mid X]\right]. $$ The outer expectation averages the conditional means using the distribution of $X$. ## Checking the values in base R ```{r} #| eval: false time <- c(400, 500, 600) mass_low <- c(.20, .50, .30) mass_high <- c(.60, .30, .10) mean_low <- sum(time * mass_low) mean_high <- sum(time * mass_high) marginal_mean <- .30 * mean_low + .70 * mean_high c(mean_low, mean_high, marginal_mean) ``` The returned values are 510, 450, and 468. ## Distinguishing a mixture change from a conditional change The marginal mean can change because the proportions of the conditions change, even when the conditional distributions remain fixed. If the share of high predictability contexts changes, the marginal mean changes even when both conditional PMFs remain fixed. Report conditional means and the condition mixture separately. A marginal difference need not imply that the within condition response process changed. ## Check your understanding 1. Recalculate the marginal mean if high and low predictability are equally probable. 2. Does changing the probability of high predictability alter either conditional mean? 3. What does the function $m(x)$ return? 4. Explain the two averaging stages in iterated expectation. Conditional expectation compares response centers across conditions. The [next page](covariance.qmd) summarizes whether paired deviations of two numerical variables tend to share a direction.