“The difficulty in constructing a Bayesian hypothesis test arises from the requirement to specify an alternative hypothesis.”

**V**ale Johnson published (and arXived) a paper in the *Annals of Statistics* on uniformly most powerful Bayesian tests. This is in line with earlier writings of Vale on the topic and good quality mathematical statistics, but I cannot really buy the arguments contained in the paper as being compatible with (my view of) Bayesian tests. A “uniformly most powerful Bayesian test” (acronymed as UMBT) is defined as

“UMPBTs provide a new form of default, nonsubjective Bayesian tests in which the alternative hypothesis is determined so as to maximize the probability that a Bayes factor exceeds a specified threshold”

which means selecting *the prior* under the alternative so that the *frequentist* probability of the Bayes factor exceeding the threshold is maximal *for all* values of the parameter. This does not sound very Bayesian to me indeed, due to this averaging over all possible values of the observations **x**** and** comparing the probabilities for all values of the parameter

**rather than integrating against a prior or posterior**

*θ***selecting the prior under the alternative with the sole purpose of favouring the alternative, meaning its further use**

*and**when*the null is rejected is not considered at all

**catering to non-Bayesian theories, i.e. trying to sell Bayesian tools as supplementing**

*and**p*-values and arguing the method is objective because the solution satisfies a frequentist coverage (at best, this maximisation of the rejection probability reminds me of minimaxity, except there is no clear and generic notion of minimaxity in hypothesis testing).