$.$ In classification, we typically want to minimize the empirical risk on the training set. This

can be represented as minimizing a $$loss function$$ applied to our training data where here

the loss function corresponds to the $0/1$ loss: $($f$($x$),$ y$)=1\{$f$($x$)=$ y$\}.$ In this case the

empirical risk of any f is simply given by

Rbn$($f$)=$ Xn

i$=1$

$($f$($xi$),$ yi$).$

In this problem we will consider the effect of instead minimizing an asymmetric loss function:

$\backslash$alpha $,\backslash$beta $($f$($x$),$ y$)=\backslash$alpha $1\{$f$($x$)=1,$ y $=0\}+\backslash$beta $1\{$f$($x$)=0,$ y $=1\}.$

Under this loss function, the two types of error receive different weights, determined by

$\backslash$alpha $,\backslash$beta $>0.$

$2$

$($a$)$ Determine the Bayes optimal classifier for this loss function, i$.$e$.,$ assuming that the

distribution of $($X$,$ Y $)$ is known, what is the classifier that minimizes the expected loss

E$[\backslash$alpha $,\backslash$beta $($f$($X$),$ Y $)]$ where $\backslash$alpha $,\backslash$beta $>0.$

$($b$)$ Suppose that the class y $=0$ is extremely uncommon $($i$.$e$.,$ P$[$Y $=0]$ is small$).$ This

means that the classifier f$($x$)=1$ for all x will have small risk$/$probability of error.

We could try to put the two classes on even footing by considering the modified risk

function:

Re$($f$)=$ P$[$f$($X$)=1|$Y $=0]+$ P$[$f$($X$)=0|$Y $=1].$

Show that minimizing Re$($f$)$ is equivalent to choosing a certain $\backslash$alpha $,\backslash$beta and minimizing

E $\backslash$alpha $,\backslash$beta $($f$($X$),$ Y $)$ for this specific $\backslash$alpha $,\backslash$beta $.$