From 7c7f3050128c49c8ec5245e51f920602d84452ce Mon Sep 17 00:00:00 2001 From: Claude Date: Thu, 22 Oct 2026 22:08:56 +0000 Subject: [PATCH 1/2] Bump latex-macros to 92f54bf and write prediction errors with \prederrf Co-Authored-By: Claude Sonnet 5.5 Claude-Session: https://claude.ai/code/session_01Lup9bgjrsSr4RGUfSDfGjB --- _subfiles/_def-prediction-error.qmd | 4 ++-- _subfiles/_def-squared-error-risk.qmd | 2 +- _subfiles/_lem-cross-term-zero.qmd | 14 +++++++------- _subfiles/_lem-prediction-error-split.qmd | 14 +++++++------- _subfiles/_thm-risk-decomposition.qmd | 8 ++++---- latex-macros | 2 +- 6 files changed, 22 insertions(+), 22 deletions(-) diff --git a/_subfiles/_def-prediction-error.qmd b/_subfiles/_def-prediction-error.qmd index edebe4da2..2e931a026 100644 --- a/_subfiles/_def-prediction-error.qmd +++ b/_subfiles/_def-prediction-error.qmd @@ -6,7 +6,7 @@ the observed [outcome](#def-outcome) $\Outvar$ of a [new observation](#def-new-o not the [conditional mean](#def-cond-expectation-function) of that outcome. The **prediction error** of $\predfn$ is the prediction minus that outcome: -$$\esterr{\predfnf{\Predvar}}{\Outvar} \eqdef \predfnf{\Predvar} - \Outvar$$ +$$\prederrf{\predfn} \eqdef \predfnf{\Predvar} - \Outvar$$ ::: @@ -17,7 +17,7 @@ $$\esterr{\predfnf{\Predvar}}{\Outvar} \eqdef \predfnf{\Predvar} - \Outvar$$ In @exr-best-predictor, consider the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$. -Its [prediction error](#def-prediction-error) is $\esterr{\predfn_1\paren{\Predvar}}{\Outvar} = 3 - \Outvar$. +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/_def-squared-error-risk.qmd b/_subfiles/_def-squared-error-risk.qmd index 3ddd1cff8..c862c7d69 100644 --- a/_subfiles/_def-squared-error-risk.qmd +++ b/_subfiles/_def-squared-error-risk.qmd @@ -5,7 +5,7 @@ The **squared-error risk** of a prediction function $\predfnf{\Predvar}$, also called its **expected prediction error** [@hastie2009elements, p. 18], is the expected square of its [prediction error](#def-prediction-error): -$$\popriskf{\predfn} \eqdef \E{\sqf{\esterr{\predfnf{\Predvar}}{\Outvar}}}$$ +$$\popriskf{\predfn} \eqdef \E{\sqf{\prederrf{\predfn}}}$$ It is the [risk](#def-risk) under [squared error loss](#def-squared-error-loss). diff --git a/_subfiles/_lem-cross-term-zero.qmd b/_subfiles/_lem-cross-term-zero.qmd index e9fefa41d..fc5ffdda9 100644 --- a/_subfiles/_lem-cross-term-zero.qmd +++ b/_subfiles/_lem-cross-term-zero.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) $\predfnf{\Predvar}$ with $\E{\predfnf{\Predvar}^2} < \infty$: -$$\E{\esterr{\truemeanf{\Predvar}}{\Outvar} \cd \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}} = 0$$ +$$\E{\prederrf{\truemean} \cd \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}} = 0$$ ::: @@ -15,11 +15,11 @@ by @thm-cond-pull-out it factors out of the [conditional expectation](#def-cond- $$ \ba -\E{\esterr{\truemeanf{\Predvar}}{\Outvar}\paren{\predfnf{\Predvar} - \truemeanf{\Predvar}} \mid \Predvar} -&= \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}} \cd \E{\esterr{\truemeanf{\Predvar}}{\Outvar} \mid \Predvar} +\E{\prederrf{\truemean}\paren{\predfnf{\Predvar} - \truemeanf{\Predvar}} \mid \Predvar} +&= \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}} \cd \E{\prederrf{\truemean} \mid \Predvar} && \text{(a function of } \Predvar \text{ factors out)} \\ &= \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}} \cd \E{\truemeanf{\Predvar} - \Outvar \mid \Predvar} -&& \text{(definition of } \esterr{\truemeanf{\Predvar}}{\Outvar} \text{)} \\ +&& \text{(definition of } \prederrf{\truemean} \text{)} \\ &= \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}} \cd \paren{\E{\truemeanf{\Predvar} \mid \Predvar} - \E{\Outvar \mid \Predvar}} && \text{(linearity of conditional expectation)} \\ &= \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}} \cd \paren{\truemeanf{\Predvar} - \truemeanf{\Predvar}} @@ -33,8 +33,8 @@ By the law of iterated expectations (@thm-lie): $$ \ba -\E{\esterr{\truemeanf{\Predvar}}{\Outvar}\paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}} -&= \E{\E{\esterr{\truemeanf{\Predvar}}{\Outvar}\paren{\predfnf{\Predvar} - \truemeanf{\Predvar}} \mid \Predvar}} +\E{\prederrf{\truemean}\paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}} +&= \E{\E{\prederrf{\truemean}\paren{\predfnf{\Predvar} - \truemeanf{\Predvar}} \mid \Predvar}} && \text{(law of iterated expectations)} \\ &= \E{0} && \text{(the conditional expectation is } 0 \text{)} \\ @@ -52,7 +52,7 @@ $$ In @exr-best-predictor, take the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$. When $\Predvar = 0$, $\truemeanf{0} = 1$, so $\predfn_1\paren{0} - \truemeanf{0} = 2$, -and $\esterr{\truemeanf{\Predvar}}{\Outvar} = 1 - \Outvar$ is $1$ or $-1$ with equal [probability](#def-probability), +and $\prederrf{\truemean} = 1 - \Outvar$ is $1$ or $-1$ with equal [probability](#def-probability), so the product has conditional mean $0$. The same cancellation happens when $\Predvar = 1$, so the expectation of the product is $0$. diff --git a/_subfiles/_lem-prediction-error-split.qmd b/_subfiles/_lem-prediction-error-split.qmd index d90805d31..036d8aafa 100644 --- a/_subfiles/_lem-prediction-error-split.qmd +++ b/_subfiles/_lem-prediction-error-split.qmd @@ -5,7 +5,7 @@ Let $\truemeanf{\Predvar} \eqdef \E{\Outvar \mid \Predvar}$ be the [conditional For any [prediction function](#def-prediction) $\predfnf{\Predvar}$, the [prediction error](#def-prediction-error) splits as: -$$\esterr{\predfnf{\Predvar}}{\Outvar} = \esterr{\truemeanf{\Predvar}}{\Outvar} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}$$ +$$\prederrf{\predfn} = \prederrf{\truemean} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}$$ ::: @@ -13,13 +13,13 @@ $$\esterr{\predfnf{\Predvar}}{\Outvar} = \esterr{\truemeanf{\Predvar}}{\Outvar} $$ \ba -\esterr{\predfnf{\Predvar}}{\Outvar} +\prederrf{\predfn} &= \predfnf{\Predvar} - \Outvar -&& \text{(definition of } \esterr{\predfnf{\Predvar}}{\Outvar} \text{)} \\ +&& \text{(definition of } \prederrf{\predfn} \text{)} \\ &= \paren{\truemeanf{\Predvar} - \Outvar} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}} && \text{(add and subtract } \truemeanf{\Predvar} \text{)} \\ -&= \esterr{\truemeanf{\Predvar}}{\Outvar} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}} -&& \text{(definition of } \esterr{\truemeanf{\Predvar}}{\Outvar} \text{)} +&= \prederrf{\truemean} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}} +&& \text{(definition of } \prederrf{\truemean} \text{)} \ea $$ @@ -32,9 +32,9 @@ $$ In @exr-best-predictor, when $\Predvar = 0$ and $\Outvar = 0$, -the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$ has [prediction error](#def-prediction-error) $\esterr{\predfn_1\paren{\Predvar}}{\Outvar} = 3$. +the constant [prediction](#def-prediction) $\predfn_1\paren{\Predvar} = 3$ has [prediction error](#def-prediction-error) $\prederrf{\predfn_1} = 3$. The conditional mean is $\truemeanf{0} = 1$, -so $\esterr{\truemeanf{\Predvar}}{\Outvar} = 1$ and $\predfn_1\paren{0} - \truemeanf{0} = 2$, +so $\prederrf{\truemean} = 1$ and $\predfn_1\paren{0} - \truemeanf{0} = 2$, and $3 = 1 + 2$. ::: diff --git a/_subfiles/_thm-risk-decomposition.qmd b/_subfiles/_thm-risk-decomposition.qmd index ea3170393..356589b62 100644 --- a/_subfiles/_thm-risk-decomposition.qmd +++ b/_subfiles/_thm-risk-decomposition.qmd @@ -19,13 +19,13 @@ and @thm-linearity-expectation: $$ \ba \popriskf{\predfn} -&= \E{\sqf{\esterr{\predfnf{\Predvar}}{\Outvar}}} +&= \E{\sqf{\prederrf{\predfn}}} && \text{(definition of the squared-error risk)} \\ -&= \E{\sqf{\esterr{\truemeanf{\Predvar}}{\Outvar} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}}} +&= \E{\sqf{\prederrf{\truemean} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}}} && \text{(the prediction error splits at } \truemean \text{)} \\ -&= \E{\sqf{\esterr{\truemeanf{\Predvar}}{\Outvar}}} + 2 \cd \E{\esterr{\truemeanf{\Predvar}}{\Outvar}\paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}} + \E{\sqf{\predfnf{\Predvar} - \truemeanf{\Predvar}}} +&= \E{\sqf{\prederrf{\truemean}}} + 2 \cd \E{\prederrf{\truemean}\paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}} + \E{\sqf{\predfnf{\Predvar} - \truemeanf{\Predvar}}} && \text{(expand the square; linearity of expectation)} \\ -&= \E{\sqf{\esterr{\truemeanf{\Predvar}}{\Outvar}}} + 2 \cd 0 + \E{\sqf{\predfnf{\Predvar} - \truemeanf{\Predvar}}} +&= \E{\sqf{\prederrf{\truemean}}} + 2 \cd 0 + \E{\sqf{\predfnf{\Predvar} - \truemeanf{\Predvar}}} && \text{(the cross term is } 0 \text{)} \\ &= \popriskf{\truemean} + \E{\sqf{\predfnf{\Predvar} - \truemeanf{\Predvar}}} && \text{(definition of the squared-error risk of } \truemean \text{)} diff --git a/latex-macros b/latex-macros index 5f14c4591..92f54bf4f 160000 --- a/latex-macros +++ b/latex-macros @@ -1 +1 @@ -Subproject commit 5f14c45912a0a8dcc88af70da58c9fcb96088f7e +Subproject commit 92f54bf4f57975ec32100896b57b1920cdc0c755 From 595ed43693d1a9c63593d190953f63c3eec7a4dd Mon Sep 17 00:00:00 2001 From: Claude Date: Thu, 22 Oct 2026 22:44:20 +0000 Subject: [PATCH 2/2] Break the risk-decomposition expansion into one term per line and use devof for the gap Co-Authored-By: Claude Sonnet 5.5 Claude-Session: https://claude.ai/code/session_01Lup9bgjrsSr4RGUfSDfGjB --- _subfiles/_thm-risk-decomposition.qmd | 13 +++++++++---- 1 file changed, 9 insertions(+), 4 deletions(-) diff --git a/_subfiles/_thm-risk-decomposition.qmd b/_subfiles/_thm-risk-decomposition.qmd index 356589b62..247563295 100644 --- a/_subfiles/_thm-risk-decomposition.qmd +++ b/_subfiles/_thm-risk-decomposition.qmd @@ -21,13 +21,18 @@ $$ \popriskf{\predfn} &= \E{\sqf{\prederrf{\predfn}}} && \text{(definition of the squared-error risk)} \\ -&= \E{\sqf{\prederrf{\truemean} + \paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}}} +&= \E{\sqf{\prederrf{\truemean} + \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}}} && \text{(the prediction error splits at } \truemean \text{)} \\ -&= \E{\sqf{\prederrf{\truemean}}} + 2 \cd \E{\prederrf{\truemean}\paren{\predfnf{\Predvar} - \truemeanf{\Predvar}}} + \E{\sqf{\predfnf{\Predvar} - \truemeanf{\Predvar}}} +&= \E{\sqf{\prederrf{\truemean}}} ++ 2 \cd \E{\prederrf{\truemean} \cd \devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}} ++ \E{\sqf{\devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}}} && \text{(expand the square; linearity of expectation)} \\ -&= \E{\sqf{\prederrf{\truemean}}} + 2 \cd 0 + \E{\sqf{\predfnf{\Predvar} - \truemeanf{\Predvar}}} +&= \E{\sqf{\prederrf{\truemean}}} ++ 2 \cd 0 ++ \E{\sqf{\devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}}} && \text{(the cross term is } 0 \text{)} \\ -&= \popriskf{\truemean} + \E{\sqf{\predfnf{\Predvar} - \truemeanf{\Predvar}}} +&= \popriskf{\truemean} ++ \E{\sqf{\devof{\predfnf{\Predvar}}{\truemeanf{\Predvar}}}} && \text{(definition of the squared-error risk of } \truemean \text{)} \ea $$