  Finite mixture models are statistical models which appear in many problems in
statistics and machine learning. In such models it is assumed that data are
drawn from random probability measures, called mixture components, which are
themselves drawn from a probability measure P over probability measures. When
estimating mixture models, it is common to make assumptions on the mixture
components, such as parametric assumptions. In this paper, we make no
assumption on the mixture components, and instead assume that observations from
the mixture model are grouped, such that observations in the same group are
known to be drawn from the same component. We show that any mixture of m
probability measures can be uniquely identified provided there are 2m-1
observations per group. Moreover we show that, for any m, there exists a
mixture of m probability measures that cannot be uniquely identified when
groups have 2m-2 observations. Our results hold for any sample space with more
than one element.
