Identifiability problems have previously precluded a general approach to testing the hypothesis of a "pure" distribution against the alternative of a mixture of distributions. Three types of identifiability are defined, and it is shown that $B$-identifiability allows a Bayesian solution to the testing problem. First, an equivalence relation is defined over parametrizations of probability functions. Then the projection onto the quotient space is shown to give a $B$-identifiable parametrization. Bayesian inference proceeds using the Bayes factor as a "test" criterion.
"Mixtures of Distributions: A Topological Approach." Ann. Statist. 16 (4) 1623 - 1634, December, 1988. https://doi.org/10.1214/aos/1176351057