Define "prediction function" separately from "prediction"

_subfiles/_def-prediction-error.qmd

@@ -1,24 +1,11 @@
 :::{#def-prediction-error}
 #### Prediction error
 
-A [prediction function](#def-prediction) $\predfnf{\Predvar}$ has as its [estimand](https://morrison-lab.github.io/sds/estimation.html#def-estimand)
+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:
 
 $$\prederrf{\predfn} \eqdef \predfnf{\Predvar} - \Outvar$$
 
 :::
-
-{{< slidebreak >}}
-
-:::{#exm-prediction-error}
-#### Prediction error in the best-predictor exercise
-
-In @exr-best-predictor,
-consider the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$.
-Its [prediction error](#def-prediction-error) is $\prederrf{\predfn_1} = 3 - \Outvar$.
-This is $3$ when $\Outvar = 0$.
-It is $-3$ when $\Outvar = 6$.
-
-:::

_subfiles/_def-prediction-function.qmd

@@ -0,0 +1,7 @@
+:::{#def-prediction-function}
+#### Prediction function
+
+A **prediction function** $\predfn$ is a [function](https://morrison-lab.github.io/mds/functions.html#def-function) that takes a value $\predvar$ of the [input](#def-input)
+and returns a [prediction](#def-prediction) $\predfnf{\predvar}$ of the [outcome](#def-outcome).
+
+:::

_subfiles/_def-reducible-risk.qmd

@@ -1,11 +1,10 @@
 :::{#def-reducible-risk}
 #### Reducible risk
 
-In the squared-error risk decomposition of @thm-risk-decomposition,
-the excess risk from choosing the prediction function $\predfn$ instead of the conditional mean,
+The **reducible risk** of a [prediction function](#def-prediction-function) $\predfn$
+is the [expectation](#def-expectation) of the squared difference
+between $\predfnf{\Predvar}$ and the [conditional mean](#def-cond-expectation-function) $\truemeanf{\Predvar}$ of the [outcome](#def-outcome):
 
-$$\E{\sqf{\E{\Outvar \mid \Predvar} - \predfnf{\Predvar}}} \ge 0$$
-
-is the **reducible risk**.
+$$\E{\sqf{\predfnf{\Predvar} - \truemeanf{\Predvar}}}$$
 
 :::

_subfiles/_def-squared-error-risk.qmd

@@ -1,21 +1,10 @@
 :::{#def-squared-error-risk}
 #### Squared-error risk (expected prediction error)
 
-The **squared-error risk** of a prediction function $\predfnf{\Predvar}$,
+The **squared-error risk** of a [prediction function](#def-prediction-function) $\predfn$,
 also called its **expected prediction error** [@hastie2009elements, p. 18],
 is the expected square of its [prediction error](#def-prediction-error):
 
 $$\popriskf{\predfn} \eqdef \E{\sqf{\prederrf{\predfn}}}$$
 
 :::
-
-{{< slidebreak >}}
-
-:::{#exm-squared-error-risk}
-#### Squared-error risk in the best-predictor exercise
-
-In @exr-best-predictor,
-consider the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$.
-Its [squared-error risk](#def-squared-error-risk) is $\popriskf{\predfn_1} = 5$.
-
-:::

_subfiles/_exm-cross-term-zero.qmd

@@ -0,0 +1,11 @@
+:::{#exm-cross-term-zero}
+#### The vanishing cross term in the best-predictor exercise
+
+In @exr-best-predictor, take the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$.
+When $\Predvar = 0$, $\truemeanf{0} = 1$, so $\predfn_1\paren{0} - \truemeanf{0} = 2$,
+and $\prederrf{\truemean} = 1 - \Outvar$ is $1$ or $-1$ with equal [probability](#def-probability),
+so the product has conditional mean $0$.
+The same cancellation happens when $\Predvar = 1$,
+so the expectation of the product is $0$.
+
+:::

_subfiles/_exm-prediction-error.qmd

@@ -0,0 +1,10 @@
+:::{#exm-prediction-error}
+#### Prediction error in the best-predictor exercise
+
+In @exr-best-predictor,
+consider the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$.
+Its [prediction error](#def-prediction-error) is $\prederrf{\predfn_1} = 3 - \Outvar$.
+This is $3$ when $\Outvar = 0$.
+It is $-3$ when $\Outvar = 6$.
+
+:::

_subfiles/_exm-prediction-function.qmd

@@ -0,0 +1,8 @@
+:::{#exm-prediction-function}
+#### A constant prediction function
+
+In @exr-best-predictor,
+the [function](https://morrison-lab.github.io/mds/functions.html#def-function) $\predfn_1\paren{\predvar} = 3$ is a [prediction function](#def-prediction-function):
+it returns the [prediction](#def-prediction) $3$ for every value of the [input](#def-input).
+
+:::

_subfiles/_exm-reducible-risk.qmd

@@ -1,8 +1,8 @@
 :::{#exm-reducible-risk}
 #### Reducible risk in the best-predictor exercise
 
 In @exr-best-predictor,
-the constant prediction $\predfn_1\paren{\Predvar} = 3$ has reducible risk $5 - 1 = 4$,
-and the prediction $\predfn_2\paren{\Predvar} = 4\Predvar$ has reducible risk $2 - 1 = 1$.
+the constant [prediction function](#def-prediction-function) $\predfn_1$, with $\predfn_1\paren{\Predvar} = 3$, has [reducible risk](#def-reducible-risk) $5 - 1 = 4$,
+and the prediction function $\predfn_2$, with $\predfn_2\paren{\Predvar} = 4\Predvar$, has reducible risk $2 - 1 = 1$.
 
 :::

_subfiles/_exm-risk-decomposition.qmd

@@ -0,0 +1,9 @@
+:::{#exm-risk-decomposition}
+#### The risk decomposition in the best-predictor exercise
+
+In @exr-best-predictor,
+the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$ has [squared-error risk](#def-squared-error-risk) $\popriskf{\predfn_1} = 5$,
+which @thm-risk-decomposition splits as $5 = 1 + 4$:
+the risk $1$ of the [conditional expectation function](#def-cond-expectation-function) plus the excess $4$ from using $\predfn_1$.
+
+:::

_subfiles/_exm-squared-error-risk.qmd

@@ -0,0 +1,8 @@
+:::{#exm-squared-error-risk}
+#### Squared-error risk in the best-predictor exercise
+
+In @exr-best-predictor,
+consider the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$.
+Its [squared-error risk](#def-squared-error-risk) is $\popriskf{\predfn_1} = 5$.
+
+:::

_subfiles/_lem-cross-term-zero.qmd

@@ -2,7 +2,7 @@
 #### The conditional mean's error is uncorrelated with any function of the input
 
 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) $\predfnf{\Predvar}$ with $\E{\predfnf{\Predvar}^2} < \infty$:
+For any [prediction function](#def-prediction-function) $\predfn$ with $\E{\predfnf{\Predvar}^2} < \infty$:
 
 $$\E{\prederrf{\truemean} \cd \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}} = 0$$
 
@@ -44,17 +44,3 @@ $$
 $$
 
 :::
-
-{{< slidebreak >}}
-
-:::{#exm-cross-term-zero}
-#### The vanishing cross term in the best-predictor exercise
-
-In @exr-best-predictor, take the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$.
-When $\Predvar = 0$, $\truemeanf{0} = 1$, so $\predfn_1\paren{0} - \truemeanf{0} = 2$,
-and $\prederrf{\truemean} = 1 - \Outvar$ is $1$ or $-1$ with equal [probability](#def-probability),
-so the product has conditional mean $0$.
-The same cancellation happens when $\Predvar = 1$,
-so the expectation of the product is $0$.
-
-:::

_subfiles/_rem-reducible-risk-notes.qmd

@@ -8,5 +8,6 @@ $$\E{\sqf{\Outvar - \predfnf{\Predvar}}} = \E{\sqf{\Outvar - \E{\Outvar \mid \Pr
 shows that the total squared-error risk is the sum of the
 residual risk (@def-irreducible-risk)
 and the reducible risk (@def-reducible-risk).
+@james2021islr2e [Section 2.1.1, p. 18] calls the squared difference between a [prediction function](#def-prediction-function) and the [conditional mean](#def-cond-expectation-function) the "reducible error".
 
 :::

_subfiles/_thm-best-predictor.qmd

@@ -3,8 +3,8 @@
 
 Let $\Predvar$ and $\Outvar$ be [jointly distributed](random-variables.qmd#def-jointly-distributed) random variables with $\E{\Outvar^2} < \infty$.
 Let $\truemeanf{\Predvar} \eqdef \E{\Outvar \mid \Predvar}$ be the [conditional expectation function](#def-cond-expectation-function) of $\Outvar$ given $\Predvar$.
-Write $\bestpredfn \eqdef \truemean$ for the prediction function that uses it.
-Then for any [prediction function](#def-prediction) $\predfnf{\Predvar}$ with $\E{\predfnf{\Predvar}^2} < \infty$:
+Write $\bestpredfn \eqdef \truemean$ for the [prediction function](#def-prediction-function) that uses it.
+Then for any prediction function $\predfn$ with $\E{\predfnf{\Predvar}^2} < \infty$:
 
 $$\popriskf{\predfn} \ge \popriskf{\bestpredfn}$$
 

_subfiles/_thm-risk-decomposition.qmd

@@ -2,7 +2,7 @@
 #### Squared-error risk decomposition
 
 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) $\predfnf{\Predvar}$ with $\E{\predfnf{\Predvar}^2} < \infty$:
+For any [prediction function](#def-prediction-function) $\predfn$ with $\E{\predfnf{\Predvar}^2} < \infty$:
 
 $$\popriskf{\predfn} = \popriskf{\truemean} + \E{\sqf{\predfnf{\Predvar} - \truemeanf{\Predvar}}}$$
 
@@ -38,15 +38,3 @@ $$
 $$
 
 :::
-
-{{< slidebreak >}}
-
-:::{#exm-risk-decomposition}
-#### The risk decomposition in the best-predictor exercise
-
-In @exr-best-predictor,
-the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$ has [squared-error risk](#def-squared-error-risk) $\popriskf{\predfn_1} = 5$,
-which @thm-risk-decomposition splits as $5 = 1 + 4$:
-the risk $1$ of the [conditional expectation function](#def-cond-expectation-function) plus the excess $4$ from using $\predfn_1$.
-
-:::

_subfiles/cor-best-predictor-argmin/_cor-best-predictor-argmin.qmd

@@ -2,7 +2,7 @@
 #### Conditional mean as the minimizer of squared-error risk
 
 Let $\Predvar$ and $\Outvar$ be [jointly distributed](random-variables.qmd#def-jointly-distributed) random variables with $\E{\Outvar^2} < \infty$.
-Among [prediction functions](#def-prediction) $\predfnf{\Predvar}$ with $\E{\predfnf{\Predvar}^2} < \infty$,
+Among [prediction functions](#def-prediction-function) $\predfn$ with $\E{\predfnf{\Predvar}^2} < \infty$,
 the [conditional expectation function](#def-cond-expectation-function) is a [minimizer](https://morrison-lab.github.io/mds/algebra-basics.html#def-global-minimizer) of the [squared-error risk](#def-squared-error-risk):
 
 $$\E{\Outvar \mid \Predvar} \in \argmin_{\predfn} \E{\sqf{\Outvar - \predfnf{\Predvar}}},$$
@@ -13,7 +13,7 @@ where $\argmin$ is the [set of minimizers](https://morrison-lab.github.io/mds/al
 
 ::: proof
 
-By @thm-best-predictor, $\bestpredfnf{\Predvar} = \E{\Outvar \mid \Predvar}$ has [risk](#def-risk) no larger than that of any [prediction function](#def-prediction) $\predfn$,
+By @thm-best-predictor, $\bestpredfnf{\Predvar} = \E{\Outvar \mid \Predvar}$ has [risk](#def-risk) no larger than that of any [prediction function](#def-prediction-function) $\predfn$,
 so it is a [minimizer](https://morrison-lab.github.io/mds/algebra-basics.html#def-global-minimizer).
 
 :::

_subfiles/cor-best-predictor-argmin/_exm-best-predictor-argmin.qmd

@@ -1,8 +1,8 @@
 :::{#exm-best-predictor-argmin}
 #### The minimizer in the best-predictor exercise
 
 In @exr-best-predictor, the [conditional expectation function](#def-cond-expectation-function) $\bestpredfnf{\Predvar}$ has [risk](#def-risk) $1$.
-The constant prediction $\predfn_1\paren{\Predvar} = 3$ has risk $5$ and $\predfn_2\paren{\Predvar} = 4\Predvar$ has risk $2$, both larger than $1$.
+The constant [prediction function](#def-prediction-function) $\predfn_1$, with $\predfn_1\paren{\Predvar} = 3$, has risk $5$ and $\predfn_2\paren{\Predvar} = 4\Predvar$ has risk $2$, both larger than $1$.
 So $\bestpredfn$ is the [minimizer](https://morrison-lab.github.io/mds/algebra-basics.html#def-global-minimizer) among the three.
 
 :::

_subfiles/cor-best-predictor-unique/_cor-best-predictor-unique.qmd

@@ -2,7 +2,7 @@
 #### The minimizer of squared-error risk is unique up to probability-1 equality
 
 Let $\Predvar$ and $\Outvar$ be [jointly distributed](random-variables.qmd#def-jointly-distributed) random variables with $\E{\Outvar^2} < \infty$.
-Every [prediction function](#def-prediction) $\predfnf{\Predvar}$ with $\E{\predfnf{\Predvar}^2} < \infty$
+Every [prediction function](#def-prediction-function) $\predfn$ with $\E{\predfnf{\Predvar}^2} < \infty$
 that is a [minimizer](https://morrison-lab.github.io/mds/algebra-basics.html#def-global-minimizer) of the [squared-error risk](#def-squared-error-risk)
 (see @cor-best-predictor-argmin)
 satisfies:

expectation.qmd

@@ -103,6 +103,14 @@ format:
 
 {{< slidebreak >}}
 
+{{< include pds/_subfiles/_def-prediction-function.qmd >}}
+
+{{< slidebreak >}}
+
+{{< include pds/_subfiles/_exm-prediction-function.qmd >}}
+
+{{< slidebreak >}}
+
 {{< include pds/_subfiles/_def-new-observation.qmd >}}
 
 {{< slidebreak >}}
@@ -111,10 +119,18 @@ format:
 
 {{< slidebreak >}}
 
+{{< include pds/_subfiles/_exm-prediction-error.qmd >}}
+
+{{< slidebreak >}}
+
 {{< include pds/_subfiles/_def-squared-error-risk.qmd >}}
 
 {{< slidebreak >}}
 
+{{< include pds/_subfiles/_exm-squared-error-risk.qmd >}}
+
+{{< slidebreak >}}
+
 {{< include pds/_subfiles/_rem-squared-error-risk-is-risk.qmd >}}
 
 {{< slidebreak >}}
@@ -127,10 +143,18 @@ format:
 
 {{< slidebreak >}}
 
+{{< include pds/_subfiles/_exm-cross-term-zero.qmd >}}
+
+{{< slidebreak >}}
+
 {{< include pds/_subfiles/_thm-risk-decomposition.qmd >}}
 
 {{< slidebreak >}}
 
+{{< include pds/_subfiles/_exm-risk-decomposition.qmd >}}
+
+{{< slidebreak >}}
+
 {{< include pds/_subfiles/_thm-best-predictor.qmd >}}
 
 {{< slidebreak >}}