From 0d2bf104e24c0736072de98bdb0c8c2ab3551f3d Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 6 Nov 2026 04:05:35 +0000 Subject: [PATCH] Add \condpopriskf for the conditional risk at an input The risk of a model at one input, R(f | x), is the expected loss given the input; the population risk is its average over the input. Duda and Hart, Hutchinson's theory notes and ESL all use the pointwise quantity, and lds had no name or symbol for it. Co-Authored-By: Claude Sonnet 5.5 Claude-Session: https://claude.ai/code/session_019xfc9o4dP1vKjn3ZbFhuJB --- _tests/pdf-include-in-header.qmd | 2 ++ interpretations.tsv | 1 + macros.qmd | 1 + notation-choices.qmd | 5 +++++ 4 files changed, 9 insertions(+) diff --git a/_tests/pdf-include-in-header.qmd b/_tests/pdf-include-in-header.qmd index ea76203a..f8757297 100644 --- a/_tests/pdf-include-in-header.qmd +++ b/_tests/pdf-include-in-header.qmd @@ -37,6 +37,8 @@ Sum of the elements, plain, squared and with a tall argument: $\sumof{\vy}$, $\S Fitted prediction function, plain, squared and applied: $\epredfn^2$, $\epredfnf{x}^2$, $\predfnf{x}_i$, $\bestpredfnf{x}^2$. +Conditional risk, plain, squared, subscripted and with a tall argument: $\condpopriskf{\modmem}{x}$, $\condpopriskf{\modmem}{x}^2$, $\condpopriskf{\modmem}{x}_i$, $\condpopriskf{\modmem}{\frac{a}{b}}^2$. + Sample means, plain, subscripted and squared: $\smean{y}_i^2$, $\xbar^2$, $\samplemean{X}{n}^2$, $\smeandev^2$, $\smeandev_i$, $\avgleverage^2$. Sums of squares, squared and subscripted: $\RSS^2$, $\RSS_0$, $\TSS_i$. Training symbols: $\nnetparams^2$, $\iter_i$, $\layeridx^2$, $\objfn_i$, $\objfnf{\vth}^2$. diff --git a/interpretations.tsv b/interpretations.tsv index c1748622..d7fbad63 100644 --- a/interpretations.tsv +++ b/interpretations.tsv @@ -172,6 +172,7 @@ name interpretation \obsloss loss on one observation, ℓ_i (argument: the observation index) \poprisk population (true) risk symbol R: the expected loss of a model \popriskf population risk of a model, R(·) +\condpopriskf conditional (pointwise) risk of a model at an input, R(f | x): the expected loss given the input; its average over the input is the population risk \lpnorm name of an ℓ_p norm or penalty, such as ℓ_1 or ℓ_2 (argument: p) \lbound a lower bound of a set (ℓ) \ubound an upper bound of a set (u) diff --git a/macros.qmd b/macros.qmd index ba33598f..fa913fb1 100644 --- a/macros.qmd +++ b/macros.qmd @@ -182,6 +182,7 @@ \providecommand{\obsloss}[1]{\ell_{#1}} \def\poprisk{R} \providecommand{\popriskf}[1]{{\poprisk\paren{#1}}} +\providecommand{\condpopriskf}[2]{{\poprisk\paren{#1 \mid #2}}} \def\modelfam{\mathcal{H}} \def\growthfn{m_{\predfam}} \providecommand{\growthfnf}[1]{{\growthfn\paren{#1}}} diff --git a/notation-choices.qmd b/notation-choices.qmd index ad3bc61a..b1430acf 100644 --- a/notation-choices.qmd +++ b/notation-choices.qmd @@ -309,6 +309,11 @@ We write the expectation, the probability, the bias and the variance as operator The blackboard letter marks an operator that takes a random variable or an estimator, so it cannot be mistaken for an italic variable. We do not use blackboard bold for the risk, because $\reals$ is the real numbers. The risk stays an italic $\poprisk$, which is free now that the noise is $\varepsilon$. +The risk of a model at one input, the conditional risk, is the same italic letter with the input after a bar: $\condpopriskf{\modmem}{x}$. +We keep one letter for both because they are one quantity at two levels. +The population risk $\popriskf{\modmem}$ is the average of $\condpopriskf{\modmem}{\predvar}$ over the distribution of the input. +Duda and Hart make the same split, between the conditional risk and the overall risk, and *The Elements of Statistical Learning* writes the pointwise form $\mathrm{EPE}(x_0)$ beside $\mathrm{EPE}(f)$. +*An Introduction to Statistical Learning* does not separate them. The long operators $\Biast$ and $\Vart$ stay upright words. We make this exception to the Greek convention for estimands in [the estimands entry](#sec-estimand-estimator-symbols). The variance operator applies to any random variable, not only to a fitted value.