A Bayesian Behrens-Fisher solution to a problem in Taxonomy

The raw data from a sample of 75 sturgeon (45 listed as shovelnose and 30 described as Alabama sturgeon) were examined to determine if there were two distinct species in this sample. Earlier work by the authors (AAB, PDB, WMH) concluded that the Alabama and shovelnose were indistinguishable based on principal component analyses of the meristic and mensural data. In our present work we perform formal Bayesian sharp hypothesis testing on the distinct principal components from the two groups (Alabama vs. Shovelnose). The question of inference on the difference of two normal means when both the variances are unknown is the celebrated Behrens-Fisher Problem. This methodology applied in the Bayesian context leads to coherent inference on the sharp null hypothesis for the difference of the two component means. Realistic prior parameters are engaged in conjugate normal prior distributions which when combined with the sampling distributions lead to the usual prior to posterior constructs. We further derive the Bayes Factor or Weighted Likelihood Ratio (WLR) for inference on our sharp null hypotheses. The evaluation of the WLR for the prior structure requires at most one dimensional quadratures which are easily computed because the integrands are products of known density functions with familiar properties.