The uniformly post powerful (UMP) test is the [[best test ]]when the alternative hypothesis is a composite hypothesis like $H_1: \theta > \theta_0$. It is uniformly best for all values of $\theta$ of the alternate hypothesis. To find the UMP test, use the [[Neyman-Pearson Lemma]] to find the best test statistic and form (direction) of the test, using any plausible value from the alternate hypothesis (e.g., any value of $\theta$ gt; \theta_0$) in the likelihood ratio. The test statistic and form (direction) of the test that works for this value will work for all values in the parameter space covered by the alternate hypothesis. For composite hypotheses of the form $H_0: \theta = \theta_0$ vs. $H_1: \theta \ne \theta_0$, there is often no UMP, depending on whether the form (direction) of the test is consistent for all values of $\theta \ne \theta_0$. It is not always the case that there is no UMP, but often the UMP does not exist.