Inconsistency of Pitman-Yor process mixtures for the number of components

In population genetics, determining the "population structure" is an important step in the analysis of sampled data. As an illustrative example, consider the impala, a species of antelope in southern Africa. Impalas are divided into two subspecies: the common impala occupying much of the eastern half of the region, and the black-faced impala inhabiting a small area in the west. While common impalas are abundant, the number of black-faced impalas has been decimated by drought, poaching, and declining resources due to human and livestock expansion. To assist conservation efforts, Lorenzen, Arctander and Siegismund (2006) collected samples from 216 impalas, and analyzed the genetic variation between/within the two subspecies. A key part of their analysis consisted of inferring the population structure -- that is, partitioning the data into distinct populations, and in particular, determining how many such populations there are. To infer the impala population structure, Lorenzen et al. employed a widely-used tool called Structure (Pritchard, Stephens and Donnelly, 2000) which, in the simplest version, models the data as a finite mixture, with each component in the mixture corresponding to a dis-Supported in part by NSF grant DMS-1007593 and DARPA contract FA8650-11-1-715.