Ex. 15.3

Ex. 15.3

Consider the simulation model used in Figure 15.7. Binary observations are generated with probabilities

\[\begin{equation} \text{Pr}(Y=1|X) = q + (1-2q)\cdot 1\left[\sum_{j=1}^JX_j>J/2\right],\non \end{equation}\]

where \(X\sim U[0,1]^p, 0\le q\le \frac{1}{2}\), and \(J\le p\) is some predefined (even) number. Describe this probability surface, and give the Bayes error rate.

Soln. 15.3

In this data model, the true decision boundary depends on the first \(J\) variables. Specifically, it depends on the sum of \(J\) uniform variables on \([0,1]\) exceeds its mean \(J/2\). Recall the Bayes error rate (e.g., (2.23) in the text), we know the Bayes error rate is

\[\begin{equation} 1 - E\left(\max_{j\in \{0,1\}} \text{Pr}(Y=j|X)\right) = q.\non \end{equation}\]