Consider a binary hypothesis testing problem of $P_0$ vs. $P_1$ under uniform costs. Let $r(\delta,\pi)$ denote the risk line for any decision rule $\delta$ and prior $\pi$, i.e, $r(\delta,\pi)=\pi R_0(\delta)+(1-\pi)R_1(\delta)$, where $R_0(\delta)=P_F(\delta)$ and $R_1(\delta)=P_M(\delta)$ are the false alarm and miss detection probabilities.
The following is a question that I am trying to solve:
If $V(\pi)=\min_{\delta} r(\delta,\pi)$ denotes the Bayes risk for prior $\pi$ and the significance level $\alpha$ is such that $V'(0) < \alpha <1$, then show that there exists Neyman Pearson decision rules that are not Bayes rules.
Can anyone help me solve this? All I know is that we can always find Bayes rules from NP rules using randomized likelihood ratio test.