Exercise 5.3 - Reject option in classifiers

For question (a), the posterior expected loss for choosing an non-reject action $ŷ$ given data $x$ is:

where $\mathbf{\text{𝐩}}$ is the column vector encoding $p(y|x)$ and $\mathbf{\text{𝐥}}(ŷ)$ is a column vector whose elements are ${\lambda }_s$ except for the $ŷ$-th one. Thus the expected loss is $(1-p(ŷ|x))\cdot {\lambda }_s$ in this case, whose minimum is obtained by let

For the reject option, the loss is uniform ${\lambda }_s$.

Thus one should choose reject or $ŷ$ by minimizing:

If

then we readily adopt the reject option. This condition is tantamount to what is required to be prove:

For question (b), the minimum of the expected loss is:

where $ŷ$ is the most probable class. When $\frac{{\lambda }_r}{{\lambda }_s}$ is negligable, the reject option would always be chosen. When $\frac{{\lambda }_r}{{\lambda }_s}$, the reject option would never be chosen.