  In applications such as gene regulatory network analysis based on single-cell
RNA sequencing data, samples often come from a mixture of different populations
and each population has its own unique network. Available graphical models
often assume that all samples are from the same population and share the same
network. One has to first cluster the samples and use available methods to
infer the network for every cluster separately. However, this two-step
procedure ignores uncertainty in the clustering step and thus could lead to
inaccurate network estimation. Motivated by these applications, we consider the
mixture Poisson log-normal model for network inference of count data from mixed
populations. The latent precision matrices of the mixture model correspond to
the networks of different populations and can be jointly estimated by
maximizing the lasso-penalized log-likelihood. Under rather mild conditions, we
show that the mixture Poisson log-normal model is identifiable and has the
positive definite Fisher information matrix. Consistency of the maximum
lasso-penalized log-likelihood estimator is also established. To avoid the
intractable optimization of the log-likelihood, we develop an algorithm called
VMPLN based on the variational inference method. Comprehensive simulation and
real single-cell RNA sequencing data analyses demonstrate the superior
performance of VMPLN.
