We consider representations of a joint distribution law of a family of categorical randomvariables (i.e., a multivariate categorical variable) as a mixture ofindependent distribution laws (i.e. distribution laws according to whichrandom variables are mutually independent). For infinite families of random variables, wedescribe a class of mixtures with identifiable mixing measure. This class is interestingfrom a practical point of view as well, as its structure clarifies principles of selectinga “good” finite family of random variables to be used in applied research. For finitefamilies of random variables, the mixing measure is never identifiable; however, it alwayspossesses a number of identifiable invariants, which provide substantial informationregarding the distribution under consideration.
