Given that the Behrens-Fisher problem appears in several places on MSE, I allow myself to signal those posts on Cross Validated:
Bayes-Poincaré solution to k-sample tests for comparison and the Behrens-Fisher problem?
Bayes-Poincaré solution to the Behrens-Fisher problem 2: calculations for Jeffreys’ priors
I propose an alternative solution and I ask whether it is right or wrong.