Question

Problem 6.2 BDR for regression

In this problem, we will consider the Bayes decision rule for regression. Suppose we have a regression problem, where $y \in \mathbb{R}$ is the output, $x \in \mathbb{R}^d$ is the input, and we have already learned the distribution $p(y|x)$, which maps the input $x$ to a distribution of outputs $y$. The goal is to select the optimal output $y$ for a given $x$.

(d) Show that the BDR for $q \rightarrow 0$ is to select the conditional mode of $p(y|x)$.