Fixed-rule error
Definition taught in Estimating error rates, part of Statistical learning
Fixed-rule error
| Type | Definition |
| Taught in | Estimating error rates |
| Part | Statistical learning |
For a trained rule \eta held fixed, the error rate under zero–one loss is
\mathrm{Err}=\mathbb{E}_F\bigl[Q\bigl(Y,\eta(X)\bigr)\bigr],
the expectation over a new draw from the distribution F of (X,Y). The class-specific rates are the same conditional error probabilities, written so that an estimator can be built from indicators. With \pi_g=\mathbb{P}(Y=g) and I(Y,g)=1 when Y=g,
\begin{align*} \mathrm{Err}_g &=\mathbb{E}_{F\mid Y=g}\bigl[Q\bigl(Y,\eta(X)\bigr)\bigr] =\mathbb{P}\bigl(\eta(X)\ne Y\mid Y=g\bigr)\\ &=\frac{\mathbb{P}\bigl(\eta(X)\ne Y,\,Y=g\bigr)}{\mathbb{P}(Y=g)} =\frac{\mathbb{E}_F\bigl[Q\bigl(Y,\eta(X)\bigr)\,I(Y,g)\bigr]}{\pi_g}. \end{align*}
They recombine by \mathrm{Err}=\sum_g\pi_g\,\mathrm{Err}_g.
Used in
- Taught in Estimating error rates, where the derivation and examples are.
- All 7 questions on Estimating error rates
See also
Zero–one loss and risk · Expectation · Estimator · Confusion matrix · Apparent error · Leave-one-out cross-validation · -fold cross-validation · Bootstrap error and the correction