Typicality and Predictive Distributions in Discriminant Function Analysis

While discriminant function analysis is an inherently Bayesian method, researchers attempting to estimate ancestry in human skeletal samples often follow discriminant function analysis with the calculation of frequentist-based typicalities for assigning group membership. Such an approach is problematic because it fails to account for admixture and for variation in why individuals may be classified as outliers or nonmembers of particular groups. This article presents an argument and methodology for employing a fully Bayesian approach in discriminant function analysis applied to cases of ancestry estimation. The approach requires adding the calculation, or estimation, of predictive distributions as the final step in ancestry-focused discriminant analyses. The methods for a fully Bayesian multivariate discriminant analysis are illustrated using craniometrics from identified population samples within the Howells published data. The article also presents ways to visualize predictive distributions calculated in more than three dimensions, explains the limitations of typicality measures, and suggests an analytical route for future studies of ancestry and admixture based in discriminant function analysis.