Add \condpopriskf for the conditional risk at an input

_tests/pdf-include-in-header.qmd

@@ -39,6 +39,8 @@ 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: $\nnetparams^2$, $\iter_i$, $\layeridx^2$, $\objfn_i$, $\objfnf{\vth}^2$.

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)

macros.qmd

@@ -183,6 +183,7 @@
 \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}}}

notation-choices.qmd

@@ -330,6 +330,11 @@ 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.