post-hoc hypothesis testing with losses

The combination of post-hoc hypothesis testing with Neyman-Pearson paradigm as decision-theory. In particular, the significance level $\alpha$ is replaced by a loss function. This is not the only way to do post-hoc testing, but it is a way. And it is currently the most well-understood way.

Setting

We consider a set of loss functions ${\ell }_b(u,v)$ for some index set $b\in \mathcal{B}$, where $u\in \{0,1\}$ represents whether the null is true (0) or the alternative is true (1), and $v$ is an action in the action space $\mathcal{A}$ (we may also allow )