  One approach for constructing copula functions is by multiplication. Given
that products of cumulative distribution functions (CDFs) are also CDFs, an
adjustment to this multiplication will result in a copula model, as discussed
by Liebscher (J Mult Analysis, 2008). Parameterizing models via products of
CDFs has some advantages, both from the copula perspective (e.g., it is
well-defined for any dimensionality) and from general multivariate analysis
(e.g., it provides models where small dimensional marginal distributions can be
easily read-off from the parameters). Independently, Huang and Frey (J Mach
Learn Res, 2011) showed the connection between certain sparse graphical models
and products of CDFs, as well as message-passing (dynamic programming) schemes
for computing the likelihood function of such models. Such schemes allows
models to be estimated with likelihood-based methods. We discuss and
demonstrate MCMC approaches for estimating such models in a Bayesian context,
their application in copula modeling, and how message-passing can be strongly
simplified. Importantly, our view of message-passing opens up possibilities to
scaling up such methods, given that even dynamic programming is not a scalable
solution for calculating likelihood functions in many models.
