I've read in various places (see e.g. the comments on this question) that the concept of Type 1 error is incompatible with Bayesian paradigms for hypothesis testing.
Why is that exactly? I can't seem to put the different pieces and definitions of what "Bayesian hypothesis testing" actually means in order to see how Type 1 error doesn't make sense in that paradigm.
EDIT: one of the answers below seems to imply that Bayesian methods do not concern themselves with whether a hypothesis is true or false, i.e. that Bayesians only deal with assigning probabilities to hypotheses, not taking actions.
But then what about Bayes' rules (i.e. decision rules that minimize the Bayes risk)? Bayes' rules still result in rejection (or non-rejection) of the null hypothesis. So it fair to say that making any decision based on a Bayes' rule is non-Bayesian at some level? I'm clearly misunderstanding something here.