614
Brian W. Junker and Jules L. Ellis
that can be represented
with mixtures of likelihoods of independent but not identically
distributed random variables, where the data satisfy a stochastic
ordering property with respect to the mixing variable. The random
variables 
may be arbitrary real-valued random variables. We
show that the distribution of 
can be given a monotone
unidimensional latent variable representation that is useful in
the sense of Junker (1993), if and only if this distribution satisfies
conditional association (CA; Holland and Rosenbaum, 1986) and a
vanishing conditional dependence (VCD) condition, which asserts
that finite subsets of the variables in 
become
independent as we condition on a larger and larger segment of the
remaining variables in 
. It is also interesting that the
mixture representation is in a certain ordinal sense unique, when CA
and VCD hold. The characterization theorem extends and simplifies the
main result of Junker (1993), and generalizes methods of Ellis and van
den Wollenberg (1993) to a much broader class of models.
Exchangeable sequences of binary random variables also satisfy both CA and VCD, as do exchangeable sequences arising as location mixtures. In the same way that de Finetti's theorem provides a path toward justifying standard iid-mixture components in hierarchical models on the basis of our intuitions about the exchangeability of observations, this theorem justifies one-dimensional latent variable components in hierarchical models, in terms of our intuitions about positive association and redundancy between observations.