Empirical Identifiability using the Topology of Mixture Likelihood Regions

Abstract

Parameter identifiability is very important if one wishes to make inferences in a statistical model. There are two types of nonidentifiability inherent to the parameters in a finite mixture model, boundary and labelling nonidentifiability. Although parameters are not identifiable in the strict sense, there is a form of asymptotic identifiability which can provide reasonable answers when components densities are well separated, relative to the sample size. In this talk we present a detailed analysis of the topology of the likelihood region to examine the role of the two key nonidentifiabilities and asymptotic identifiability on set estimation for the mixture parameters. We show that the likelihood region has a natural partition into identifiable subsets as long as the target confidence level is chosen small enough. Any element of the partition can be used to form an identifiable unimodal confidence region for labelled parameters. However, at confidence levels that are too high, it is clear that there is no natural way to use the likelihood to resolve the identifiability problem. We also develop a fundamental theory for the existence of an identifiable partition ?in the mixture likelihood and apply this theory to better understand the role of the mixture parameters in the identifiable partition. In particular, we show that the weight parameters carry no information about construction of an identifiable unimodal partition, but order restricted inference on a univariate component parameter is equivalent to the likelihood region partition at suitable confidence levels.