  Discrete data such as counts of microbiome taxa resulting from
next-generation sequencing are routinely encountered in bioinformatics. Taxa
count data in microbiome studies are typically high-dimensional,
over-dispersed, and can only reveal relative abundance therefore being treated
as compositional. Analyzing compositional data presents many challenges because
they are restricted on a simplex. In a logistic normal multinomial model, the
relative abundance is mapped from a simplex to a latent variable that exists on
the real Euclidean space using the additive log-ratio transformation. While a
logistic normal multinomial approach brings in flexibility for modeling the
data, it comes with a heavy computational cost as the parameter estimation
typically relies on Bayesian techniques. In this paper, we develop a novel
mixture of logistic normal multinomial models for clustering microbiome data.
Additionally, we utilize an efficient framework for parameter estimation using
variational Gaussian approximations (VGA). Adopting a variational Gaussian
approximation for the posterior of the latent variable reduces the
computational overhead substantially. The proposed method is illustrated on
simulated and real datasets.
