Definition 2 (Gain function)

In Problem 3, for a point \(\theta \in Y\) and its prediction \(y \in Y\) , a gain function is defined as \(G: Y \times Y \rightarrow \mathbb{R}^+, G(\theta, y)\).

Definition 3 (MEG estimator)

In Problem 3 with Assumption 1, the maximum expected gain (MEG) estimator is defined as
\[
\newcommand{\argmax}{\mathop{\rm argmax}\limits}
\hat{y}^{(MEG)} = \argmax_{y \in Y} \sum_{\theta \in Y} G(\theta, y) p(\theta|D)
\]