From dc285d7ea31c667e339742d14d2d8952f88360d5 Mon Sep 17 00:00:00 2001
From: Claude <noreply@anthropic.com>
Date: Thu, 29 Oct 2026 22:33:00 +0000
Subject: [PATCH 1/8] Name the gap between a prediction function and the
 conditional mean

Finish the prediction-function relink in the two remarks that follow the
definition. In the prediction-error split, write the gap with the shared
deviation macro and name it as the deviation of the prediction function from
the conditional mean, the convention in the macros notation-choices page.

Co-Authored-By: Claude Sonnet 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_019xfc9nVFCSNv3sBfesyDpL
---
 _subfiles/_lem-prediction-error-split.qmd | 11 ++++++-----
 _subfiles/_rem-irreducible-risk.qmd       |  2 +-
 _subfiles/_rem-reducible-risk.qmd         |  2 +-
 3 files changed, 8 insertions(+), 7 deletions(-)

diff --git a/_subfiles/_lem-prediction-error-split.qmd b/_subfiles/_lem-prediction-error-split.qmd
index 036d8aafa..448dec682 100644
--- a/_subfiles/_lem-prediction-error-split.qmd
+++ b/_subfiles/_lem-prediction-error-split.qmd
@@ -2,10 +2,11 @@
 #### 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) $\predfnf{\Predvar}$,
+the [prediction error](#def-prediction-error) is the prediction error of $\truemean$
+plus the [deviation](#def-deviation) of $\predfnf{\Predvar}$ from $\truemeanf{\Predvar}$:
 
-$$\prederrf{\predfn} = \prederrf{\truemean} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}$$
+$$\prederrf{\predfn} = \prederrf{\truemean} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}$$
 
 :::
 
@@ -16,9 +17,9 @@ $$
 \prederrf{\predfn}
 &= \predfnf{\Predvar} - \Outvar
 && \text{(definition of } \prederrf{\predfn} \text{)} \\
-&= \paren{\truemeanf{\Predvar} - \Outvar} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}
+&= \paren{\truemeanf{\Predvar} - \Outvar} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}
 && \text{(add and subtract } \truemeanf{\Predvar} \text{)} \\
-&= \prederrf{\truemean} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}
+&= \prederrf{\truemean} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}
 && \text{(definition of } \prederrf{\truemean} \text{)}
 \ea
 $$
diff --git a/_subfiles/_rem-irreducible-risk.qmd b/_subfiles/_rem-irreducible-risk.qmd
index 821b91fd4..51281ba38 100644
--- a/_subfiles/_rem-irreducible-risk.qmd
+++ b/_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,
diff --git a/_subfiles/_rem-reducible-risk.qmd b/_subfiles/_rem-reducible-risk.qmd
index 62a5ed7d9..4a792f396 100644
--- a/_subfiles/_rem-reducible-risk.qmd
+++ b/_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

From fa0ad20cbb050d7fb8244e71d013af594888bda8 Mon Sep 17 00:00:00 2001
From: Claude <noreply@anthropic.com>
Date: Thu, 29 Oct 2026 22:34:29 +0000
Subject: [PATCH 2/8] Define the prediction error of a prediction, not of a
 prediction function

The prediction error is the prediction minus the outcome, so define it for a
prediction (a constant or g(X)) with only that equation. Move the example and
the note about the prediction function and the reference value to their own
subfiles.

Co-Authored-By: Claude Sonnet 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_019xfc9nVFCSNv3sBfesyDpL
---
 _subfiles/_def-prediction-error.qmd | 21 +++------------------
 _subfiles/_exm-prediction-error.qmd | 10 ++++++++++
 _subfiles/_rem-prediction-error.qmd | 10 ++++++++++
 expectation.qmd                     |  8 ++++++++
 4 files changed, 31 insertions(+), 18 deletions(-)
 create mode 100644 _subfiles/_exm-prediction-error.qmd
 create mode 100644 _subfiles/_rem-prediction-error.qmd

diff --git a/_subfiles/_def-prediction-error.qmd b/_subfiles/_def-prediction-error.qmd
index a75998d89..118cabef5 100644
--- a/_subfiles/_def-prediction-error.qmd
+++ b/_subfiles/_def-prediction-error.qmd
@@ -1,24 +1,9 @@
 :::{#def-prediction-error}
 #### Prediction error
 
-A [prediction function](#def-prediction-function) $\predfnf{\Predvar}$ 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:
 
-$$\prederrf{\predfn} \eqdef \predfnf{\Predvar} - \Outvar$$
-
-:::
-
-{{< 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 $\Outvar = 6$.
+$$\prederrf{\est{\Outvar}} \eqdef \est{\Outvar} - \Outvar$$
 
 :::
diff --git a/_subfiles/_exm-prediction-error.qmd b/_subfiles/_exm-prediction-error.qmd
new file mode 100644
index 000000000..944ea01bf
--- /dev/null
+++ b/_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 $\Outvar = 6$.
+
+:::
diff --git a/_subfiles/_rem-prediction-error.qmd b/_subfiles/_rem-prediction-error.qmd
new file mode 100644
index 000000000..8401e5a22
--- /dev/null
+++ b/_subfiles/_rem-prediction-error.qmd
@@ -0,0 +1,10 @@
+:::{#rem-prediction-error}
+#### The prediction error of a prediction function
+
+A [prediction function](#def-prediction-function) $\predfn$ makes the prediction $\predfnf{\Predvar}$ for the 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, $\prederrf{\predfn} = \predfnf{\Predvar} - \Outvar$.
+
+The reference value in a prediction error is the observed outcome $\Outvar$,
+not the [conditional mean](#def-cond-expectation-function) of that outcome.
+
+:::
diff --git a/expectation.qmd b/expectation.qmd
index 486ccf95c..492c675d0 100644
--- a/expectation.qmd
+++ b/expectation.qmd
@@ -119,6 +119,14 @@ format:
 
 {{< slidebreak >}}
 
+{{< include pds/_subfiles/_exm-prediction-error.qmd >}}
+
+{{< slidebreak >}}
+
+{{< include pds/_subfiles/_rem-prediction-error.qmd >}}
+
+{{< slidebreak >}}
+
 {{< include pds/_subfiles/_def-squared-error-risk.qmd >}}
 
 {{< slidebreak >}}

From 2939bf51e98c305ed1111e09458dd679ab013c17 Mon Sep 17 00:00:00 2001
From: Claude <noreply@anthropic.com>
Date: Thu, 29 Oct 2026 22:34:53 +0000
Subject: [PATCH 3/8] Link prediction and outcome in the prediction-error
 remark

Co-Authored-By: Claude Sonnet 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_019xfc9nVFCSNv3sBfesyDpL
---
 _subfiles/_rem-prediction-error.qmd | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/_subfiles/_rem-prediction-error.qmd b/_subfiles/_rem-prediction-error.qmd
index 8401e5a22..073fc3a93 100644
--- a/_subfiles/_rem-prediction-error.qmd
+++ b/_subfiles/_rem-prediction-error.qmd
@@ -1,7 +1,7 @@
 :::{#rem-prediction-error}
 #### The prediction error of a prediction function
 
-A [prediction function](#def-prediction-function) $\predfn$ makes the prediction $\predfnf{\Predvar}$ for the outcome $\Outvar$ of a [new observation](#def-new-observation) with [input](#def-input) $\Predvar$.
+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, $\prederrf{\predfn} = \predfnf{\Predvar} - \Outvar$.
 
 The reference value in a prediction error is the observed outcome $\Outvar$,

From 1d151c8d3e7f530fcf1f85702f6bd38c0382e02c Mon Sep 17 00:00:00 2001
From: Claude <noreply@anthropic.com>
Date: Thu, 29 Oct 2026 22:40:23 +0000
Subject: [PATCH 4/8] Link deviation cross-page, bridge prediction error
 remark, use devof in example

Co-Authored-By: Claude Sonnet 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_019xfc9nVFCSNv3sBfesyDpL
---
 _subfiles/_exm-prediction-error-split.qmd | 2 +-
 _subfiles/_lem-prediction-error-split.qmd | 2 +-
 _subfiles/_rem-prediction-error.qmd       | 9 +++++++--
 3 files changed, 9 insertions(+), 4 deletions(-)

diff --git a/_subfiles/_exm-prediction-error-split.qmd b/_subfiles/_exm-prediction-error-split.qmd
index 6e23a90c9..4c4284ea9 100644
--- a/_subfiles/_exm-prediction-error-split.qmd
+++ b/_subfiles/_exm-prediction-error-split.qmd
@@ -5,7 +5,7 @@ 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 $\predfn_1\paren{0} - \truemeanf{0} = 2$,
+so $\prederrf{\truemean} = 1$ and $\devof{\predfn_1\paren{0}}{\truemeanf{0}} = 2$,
 and $3 = 1 + 2$.
 
 :::
diff --git a/_subfiles/_lem-prediction-error-split.qmd b/_subfiles/_lem-prediction-error-split.qmd
index 3ee7ad22c..a67da0bd3 100644
--- a/_subfiles/_lem-prediction-error-split.qmd
+++ b/_subfiles/_lem-prediction-error-split.qmd
@@ -4,7 +4,7 @@
 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-function) $\predfnf{\Predvar}$,
 the [prediction error](#def-prediction-error) is the prediction error of $\truemean$
-plus the [deviation](#def-deviation) of $\predfnf{\Predvar}$ from $\truemeanf{\Predvar}$:
+plus the [deviation](variance-covariance.qmd#def-deviation) of $\predfnf{\Predvar}$ from $\truemeanf{\Predvar}$:
 
 $$\prederrf{\predfn} = \prederrf{\truemean} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}$$
 
diff --git a/_subfiles/_rem-prediction-error.qmd b/_subfiles/_rem-prediction-error.qmd
index 073fc3a93..25cecafdb 100644
--- a/_subfiles/_rem-prediction-error.qmd
+++ b/_subfiles/_rem-prediction-error.qmd
@@ -1,8 +1,13 @@
 :::{#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, $\prederrf{\predfn} = \predfnf{\Predvar} - \Outvar$.
+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 $\prederrf{\predfnf{\Predvar}}$,
+which 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.

From f8206c1cdcb54c3b5789e5fe382d328e9316e53b Mon Sep 17 00:00:00 2001
From: Claude <noreply@anthropic.com>
Date: Thu, 29 Oct 2026 22:40:53 +0000
Subject: [PATCH 5/8] Link prediction function in cor-best-predictor-unique to
 its definition

Co-Authored-By: Claude Sonnet 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_019xfc9nVFCSNv3sBfesyDpL
---
 .../cor-best-predictor-unique/_cor-best-predictor-unique.qmd    | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/_subfiles/cor-best-predictor-unique/_cor-best-predictor-unique.qmd b/_subfiles/cor-best-predictor-unique/_cor-best-predictor-unique.qmd
index 1db240f65..13505364e 100644
--- a/_subfiles/cor-best-predictor-unique/_cor-best-predictor-unique.qmd
+++ b/_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) $\predfnf{\Predvar}$ 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:

From e09dc84e97c211afe5227c1059aebcc95dcff4d3 Mon Sep 17 00:00:00 2001
From: Claude <noreply@anthropic.com>
Date: Sat, 31 Oct 2026 01:14:01 +0000
Subject: [PATCH 6/8] Restore the lemma's subject and cite the remark that
 introduces the prediction error notation

Co-Authored-By: Claude Sonnet 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01FWgYrE7LQ7q9Lw5qkHaNiT
---
 _subfiles/_lem-prediction-error-split.qmd | 10 ++++++----
 1 file changed, 6 insertions(+), 4 deletions(-)

diff --git a/_subfiles/_lem-prediction-error-split.qmd b/_subfiles/_lem-prediction-error-split.qmd
index a67da0bd3..94f088a12 100644
--- a/_subfiles/_lem-prediction-error-split.qmd
+++ b/_subfiles/_lem-prediction-error-split.qmd
@@ -2,8 +2,8 @@
 #### 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-function) $\predfnf{\Predvar}$,
-the [prediction error](#def-prediction-error) is the prediction error of $\truemean$
+For any [prediction function](#def-prediction-function) $\predfn$,
+the [prediction error](#def-prediction-error) of $\predfn$ is the prediction error of $\truemean$
 plus the [deviation](variance-covariance.qmd#def-deviation) of $\predfnf{\Predvar}$ from $\truemeanf{\Predvar}$:
 
 $$\prederrf{\predfn} = \prederrf{\truemean} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}$$
@@ -12,15 +12,17 @@ $$\prederrf{\predfn} = \prederrf{\truemean} + \devof{\predfnf{\Predvar}}{\trueme
 
 ::: proof
 
+The notation $\prederrf{\predfn}$ is introduced in @rem-prediction-error.
+
 $$
 \ba
 \prederrf{\predfn}
 &= \predfnf{\Predvar} - \Outvar
-&& \text{(definition of } \prederrf{\predfn} \text{)} \\
+&& \text{(prediction error notation)} \\
 &= \paren{\truemeanf{\Predvar} - \Outvar} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}
 && \text{(add and subtract } \truemeanf{\Predvar} \text{)} \\
 &= \prederrf{\truemean} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}
-&& \text{(definition of } \prederrf{\truemean} \text{)}
+&& \text{(prediction error notation)}
 \ea
 $$
 

From 096dcf8326076d72d078a2ec08011cc6f759ddda Mon Sep 17 00:00:00 2001
From: Claude <noreply@anthropic.com>
Date: Sat, 31 Oct 2026 01:25:00 +0000
Subject: [PATCH 7/8] Link prediction and conditional mean in the lemma

Co-Authored-By: Claude Sonnet 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01FWgYrE7LQ7q9Lw5qkHaNiT
---
 _subfiles/_lem-prediction-error-split.qmd | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/_subfiles/_lem-prediction-error-split.qmd b/_subfiles/_lem-prediction-error-split.qmd
index 6071823f7..a227cbd5d 100644
--- a/_subfiles/_lem-prediction-error-split.qmd
+++ b/_subfiles/_lem-prediction-error-split.qmd
@@ -4,7 +4,7 @@
 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-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 of $\predfn$ and the conditional mean
+plus the difference $\devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}$ between the [prediction](#def-prediction) of $\predfn$ and the [conditional mean](#def-cond-expectation-function)
 (the [deviation](variance-covariance.qmd#def-deviation) of the former from the latter, defined in a later chapter):
 
 $$\prederrf{\predfn} = \prederrf{\truemean} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}$$

From 121785f45b4bb6cbe9c0e751ad92a60bf9b9188a Mon Sep 17 00:00:00 2001
From: Claude <noreply@anthropic.com>
Date: Sat, 31 Oct 2026 01:31:03 +0000
Subject: [PATCH 8/8] State the lemma's difference plainly with one link per
 term; call f1 a constant prediction function

Co-Authored-By: Claude Sonnet 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01FWgYrE7LQ7q9Lw5qkHaNiT
---
 _subfiles/_exm-prediction-error-split.qmd | 2 +-
 _subfiles/_lem-prediction-error-split.qmd | 3 +--
 2 files changed, 2 insertions(+), 3 deletions(-)

diff --git a/_subfiles/_exm-prediction-error-split.qmd b/_subfiles/_exm-prediction-error-split.qmd
index 4c4284ea9..6946f6e5e 100644
--- a/_subfiles/_exm-prediction-error-split.qmd
+++ b/_subfiles/_exm-prediction-error-split.qmd
@@ -3,7 +3,7 @@
 
 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 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 $3 = 1 + 2$.
diff --git a/_subfiles/_lem-prediction-error-split.qmd b/_subfiles/_lem-prediction-error-split.qmd
index a227cbd5d..98daa71f6 100644
--- a/_subfiles/_lem-prediction-error-split.qmd
+++ b/_subfiles/_lem-prediction-error-split.qmd
@@ -4,8 +4,7 @@
 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-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) of $\predfn$ and the [conditional mean](#def-cond-expectation-function)
-(the [deviation](variance-covariance.qmd#def-deviation) of the former from the latter, defined in a later chapter):
+plus the difference $\devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}$ between the [prediction](#def-prediction) $\predfnf{\Predvar}$ and $\truemeanf{\Predvar}$:
 
 $$\prederrf{\predfn} = \prederrf{\truemean} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}$$