I could not find an answer to this question, so it's probably obvious to everyone else:
This question concerns binary hypothesis testing ($\mathcal{H}_{0}$ versus $\mathcal{H}_{1}$) against data $\mathbf{x}$. I would like to know if we can solve for a Neyman-Pearson (NP) threshold that is equivalent to a threshold that is given by a Bayes test that minimizes Risk under $0-1$ loss.
Disclaimer: I am not a mathematician or statistician.
Let us keep it simple and assume we know the PDF for $\mathbf{x}$ under both hypotheses (I will worry about GLRTs on my own). The the NP decision rule is:
\begin{equation} \cfrac{ f\left(\mathbf{x} \, \vert \, \mathcal{H}_{1} \right) } {f\left(\mathbf{x} \, \vert \, \mathcal{H}_{0} \right) } \,\, \underset{\mathcal{H}_{0}} { \overset{\mathcal{H}_{1}} {\gtrless}} \,\, t, \end{equation}
where $t$ is the threshold, $f\left(\mathbf{x} \, \vert \, \mathcal{H}_{1} \right)$ is the density for $\mathbf{x}$ under $\mathcal{H}_{1}$, and $f\left(\mathbf{x} \, \vert \, \mathcal{H}_{0} \right)$ is the density for $\mathbf{x}$ under $\mathcal{H}_{0}$. If we set threshold $t$ for a fixed false alarm probability $\text{Pr}_{FA} = \alpha$ as consistent with the NP criterion, then:
\begin{equation} t = F^{-1}(1 - \alpha \, \vert \,\mathcal{H}_{0}) \end{equation}
$F^{-1}( \mathbf{x} \, \vert \,\mathcal{H}_{0})$ is the inverse cumulative distribution for the data $\mathbf{x}$ under the null. The Bayes threshold for the same hypothesis test that minimizes risk, under $0-1$ loss:
\begin{equation} t = \cfrac{ C_{10} - C_{00} }{ C_{01} - C_{11} }\cfrac{ \text{Pr}(\mathcal{H}_{0}) }{ \text{Pr}(\mathcal{H}_{1}) } = \cfrac{ C_{10} - C_{00} }{ C_{01} - C_{11} }\cfrac{ \text{Pr}(\mathcal{H}_{0}) }{ 1- \text{Pr}(\mathcal{H}_{0}) } \end{equation}
The $C_{ij}$ are the costs. The cost $C_{ij}$ is the cost if we decide that $\mathcal{H}_{i}$ is true, but $\mathcal{H}_{j}$ is true. This is the general expression, but I'll simplify it a bit more and assume correct decisions have no cost. I then equate the thresholds (maybe you'll tell me that this is philosophically contradictory):
\begin{equation} \cfrac{ C_{10} }{ C_{01} }\cfrac{ \text{Pr}(\mathcal{H}_{0}) }{ 1- \text{Pr}(\mathcal{H}_{0}) } = F^{-1}(1 - \alpha \, \vert \,\mathcal{H}_{0}) \end{equation}
I could also operate on each side of this equation with cumulative distribution under the null, $F(\bullet \, \vert \,\mathcal{H}_{0})$, but it's just the same equation re-stated.
Why do I care about this? I would like to see if what false alarm rate I set, $\alpha$, is equivalent to a set of costs, given my priors. I just need to know if anything I have written here is B.S. for some reason. Thanks!