  Multivariate count data are commonly encountered through high-throughput
sequencing technologies in bioinformatics, text mining, or in sports analytics.
Although the Poisson distribution seems a natural fit to these count data, its
multivariate extension is computationally expensive.In most cases mutual
independence among the variables is assumed, however this fails to take into
account the correlation among the variables usually observed in the data.
Recently, mixtures of multivariate Poisson-lognormal (MPLN) models have been
used to analyze such multivariate count measurements with a dependence
structure. In the MPLN model, each count is modeled using an independent
Poisson distribution conditional on a latent multivariate Gaussian variable.
Due to this hierarchical structure, the MPLN model can account for
over-dispersion as opposed to the traditional Poisson distribution and allows
for correlation between the variables. Rather than relying on a Monte
Carlo-based estimation framework which is computationally inefficient, a fast
variational-EM based framework is used here for parameter estimation. Further,
a parsimonious family of mixtures of Poisson-lognormal distributions are
proposed by decomposing the covariance matrix and imposing constraints on these
decompositions. Utility of such models is shown using simulated and benchmark
datasets.
