Define the prediction error of a prediction; name the deviation from the conditional mean

_subfiles/_def-prediction-error.qmd

@@ -1,11 +1,9 @@
 :::{#def-prediction-error}
 #### Prediction error
 
-A [prediction function](#def-prediction-function) $\predfn$ has as its [estimand](https://morrison-lab.github.io/sds/estimation.html#def-estimand)
-the observed [outcome](#def-outcome) $\Outvar$ of a [new observation](#def-new-observation) with [input](#def-input) $\Predvar$,
-not the [conditional mean](#def-cond-expectation-function) of that outcome.
-The **prediction error** of $\predfn$ is the prediction minus that outcome:
+The **prediction error** of a [prediction](#def-prediction) $\est{\Outvar}$ of an [outcome](#def-outcome) $\Outvar$
+is the prediction minus the outcome:
 
-$$\prederrf{\predfn} \eqdef \predfnf{\Predvar} - \Outvar$$
+$$\prederrf{\est{\Outvar}} \eqdef \est{\Outvar} - \Outvar$$
 
 :::

_subfiles/_exm-prediction-error-split.qmd

@@ -0,0 +1,11 @@
+:::{#exm-prediction-error-split}
+#### The split in the best-predictor exercise
+
+In @exr-best-predictor,
+when $\Predvar = 0$ and $\Outvar = 0$,
+the constant [prediction function](#def-prediction-function) $\predfn_1$, with $\predfn_1\paren{\Predvar} = 3$, has [prediction error](#def-prediction-error) $\prederrf{\predfn_1} = 3$.
+The conditional mean is $\truemeanf{0} = 1$,
+so $\prederrf{\truemean} = 1$ and $\devof{\predfn_1\paren{0}}{\truemeanf{0}} = 2$,
+and $3 = 1 + 2$.
+
+:::

_subfiles/_lem-prediction-error-split.qmd

@@ -2,39 +2,28 @@
 #### Prediction error splits at the conditional mean
 
 Let $\truemeanf{\Predvar} \eqdef \E{\Outvar \mid \Predvar}$ be the [conditional expectation function](#def-cond-expectation-function) of $\Outvar$ given $\Predvar$.
-For any [prediction function](#def-prediction) $\predfnf{\Predvar}$,
-the [prediction error](#def-prediction-error) splits as:
+For any [prediction function](#def-prediction-function) $\predfn$,
+the [prediction error](#def-prediction-error) of $\predfn$ is the prediction error of $\truemean$
+plus the difference $\devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}$ between the [prediction](#def-prediction) $\predfnf{\Predvar}$ and $\truemeanf{\Predvar}$:
 
-$$\prederrf{\predfn} = \prederrf{\truemean} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}$$
+$$\prederrf{\predfn} = \prederrf{\truemean} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}$$
 
 :::
 
 ::: proof
 
+The notation $\prederrf{\predfn}$ is introduced in @rem-prediction-error.
+
 $$
 \ba
 \prederrf{\predfn}
 &= \predfnf{\Predvar} - \Outvar
-&& \text{(definition of } \prederrf{\predfn} \text{)} \\
-&= \paren{\truemeanf{\Predvar} - \Outvar} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}
+&& \text{(prediction error notation)} \\
+&= \paren{\truemeanf{\Predvar} - \Outvar} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}
 && \text{(add and subtract } \truemeanf{\Predvar} \text{)} \\
-&= \prederrf{\truemean} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}
-&& \text{(definition of } \prederrf{\truemean} \text{)}
+&= \prederrf{\truemean} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}
+&& \text{(prediction error notation)}
 \ea
 $$
 
 :::
-
-{{< slidebreak >}}
-
-:::{#exm-prediction-error-split}
-#### The split in the best-predictor exercise
-
-In @exr-best-predictor,
-when $\Predvar = 0$ and $\Outvar = 0$,
-the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$ has [prediction error](#def-prediction-error) $\prederrf{\predfn_1} = 3$.
-The conditional mean is $\truemeanf{0} = 1$,
-so $\prederrf{\truemean} = 1$ and $\predfn_1\paren{0} - \truemeanf{0} = 2$,
-and $3 = 1 + 2$.
-
-:::

_subfiles/_rem-irreducible-risk.qmd

@@ -4,7 +4,7 @@
 The [residual risk](#def-irreducible-risk) depends only on the [joint distribution](random-variables.qmd#def-joint-distribution) of $(\Predvar, \Outvar)$
 and not on the prediction function $\predfn$,
 so no choice of $\predfn$ can achieve a smaller risk.
-It is the minimum only among [prediction functions](#def-prediction) of the given $\Predvar$:
+It is the minimum only among [prediction functions](#def-prediction-function) of the given $\Predvar$:
 the [outcome](#def-outcome) may also depend on variables that $\Predvar$ leaves out,
 and those variables are part of the unexplained variation $\Outvar - \E{\Outvar \mid \Predvar}$.
 Measuring one of them and adding it to the variables in $\Predvar$ can lower the residual risk,

_subfiles/_rem-prediction-error.qmd

@@ -0,0 +1,15 @@
+:::{#rem-prediction-error}
+#### The prediction error of a prediction function
+
+A [prediction function](#def-prediction-function) $\predfn$ makes the [prediction](#def-prediction) $\predfnf{\Predvar}$
+for the [outcome](#def-outcome) $\Outvar$ of a [new observation](#def-new-observation) with [input](#def-input) $\Predvar$.
+We write $\prederrf{\predfn}$ for the [prediction error](#def-prediction-error) of that prediction,
+meaning $\prederrf{\predfnf{\Predvar}}$.
+It is
+
+$$\prederrf{\predfn} = \predfnf{\Predvar} - \Outvar$$
+
+The reference value in a prediction error is the observed outcome $\Outvar$,
+not the [conditional mean](#def-cond-expectation-function) of that outcome.
+
+:::

_subfiles/_rem-reducible-risk.qmd

@@ -1,7 +1,7 @@
 ::: {#rem-reducible-risk}
 #### When the reducible risk is zero
 
-The [reducible risk](#def-reducible-risk) is zero if and only if the [prediction function](#def-prediction) $\predfn$ equals the [conditional expectation function](#def-cond-expectation-function) with [probability](probability-basics.qmd#def-probability) 1.
+The [reducible risk](#def-reducible-risk) is zero if and only if the [prediction function](#def-prediction-function) $\predfn$ equals the [conditional expectation function](#def-cond-expectation-function) with [probability](probability-basics.qmd#def-probability) 1.
 
 $$
 \E{\sqf{\E{\Outvar \mid \Predvar} - \predfnf{\Predvar}}} = 0

expectation.qmd

@@ -123,6 +123,10 @@ format:
 
 {{< slidebreak >}}
 
+{{< include pds/_subfiles/_rem-prediction-error.qmd >}}
+
+{{< slidebreak >}}
+
 {{< include pds/_subfiles/_def-squared-error-risk.qmd >}}
 
 {{< slidebreak >}}
@@ -139,6 +143,10 @@ format:
 
 {{< slidebreak >}}
 
+{{< include pds/_subfiles/_exm-prediction-error-split.qmd >}}
+
+{{< slidebreak >}}
+
 {{< include pds/_subfiles/_lem-cross-term-zero.qmd >}}
 
 {{< slidebreak >}}