  For in vivo research experiments with small sample sizes and available
historical data, we propose a sequential Bayesian method for the Behrens-Fisher
problem. We consider it as a model choice question with two models in
competition: one for which the two expectations are equal and one for which
they are different. The choice between the two models is performed through a
Bayesian analysis, based on a robust choice of combined objective and
subjective priors, set on the parameters space and on the models space. Three
steps are necessary to evaluate the posterior probability of each model using
two historical datasets similar to the one of interest. Starting from the
Jeffreys prior, a posterior using a first historical dataset is deduced and
allows to calibrate the Normal-Gamma informative priors for the second
historical dataset analysis, in addition to a uniform prior on the model space.
From this second step, a new posterior on the parameter space and the models
space can be used as the objective informative prior for the last Bayesian
analysis. Bayesian and frequentist methods have been compared on simulated and
real data. In accordance with FDA recommendations, control of type I and type
II error rates has been evaluated. The proposed method controls them even if
the historical experiments are not completely similar to the one of interest.
