We consider representations of a joint distribution law of a family of categorical random
variables (i.e., a multivariate categorical variable) as a mixture of
independent distribution laws (i.e. distribution laws according to which
random variables are mutually independent). For infinite families of random variables, we
describe a class of mixtures with identifiable mixing measure. This class is interesting
from a practical point of view as well, as its structure clarifies principles of selecting
a “good” finite family of random variables to be used in applied research. For finite
families of random variables, the mixing measure is never identifiable; however, it always
possesses a number of identifiable invariants, which provide substantial information
regarding the distribution under consideration.