Mixture identifiability

Simultaneously applying a permutation to the mixture weights and mixture components leaves $f$ unchanged. Thus identifiability of named components requires a convention, such as ordering distinct component means; label switching describes the associated inferential issue.^[ref]

Even after choosing label names, identical mixture components reveal only their combined mixture weight, and an inactive mixture component has undetermined statistical parameters. Adding mixture components increases flexibility but can cause overfitting; judge predictions using cross-validation, rather than interpreting every fitted component as a discovered population.^[ref]