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Multivariate count data are commonly encountered through high‐throughput sequencing technologies in bioinformatics, text mining, or 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. Owing 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 expectation–maximization (EM)‐based framework is used here for parameter estimation. Further, a family of parsimonious mixtures of Poisson‐lognormal distributions is proposed by decomposing the covariance matrix and imposing constraints on these decompositions. Utility of such models is shown using simulated and benchmark datasets. [ABSTRACT FROM AUTHOR]