Variational Approximation-Based Model Selection for Microbial Network Inference

Microbial associations are characterized by both direct and indirect interactions between the constituent taxa in a microbial community, and play an important role in determining the structure, organization, and function of the community. Microbial associations can be represented using a weighted graph (microbial network), whose nodes represent taxa and edges represent pairwise associations. A microbial network is typically inferred from a sample-taxa matrix that is obtained by sequencing multiple biological samples and identifying the taxa counts in each sample. However, it is known that microbial associations are impacted by environmental and/or host factors. Thus, a sample-taxa matrix generated in a microbiome study involving a wide range of values for the environmental and/or clinical metadata variables may in fact be associated with more than one microbial network. In this study, we consider the problem of inferring multiple microbial networks from a given sample-taxa count matrix. Each sample is a count vector assumed to be generated by a mixture model consisting of component distributions that are multivariate Poisson log-normal. We present a variational expectation maximization algorithm for the model selection problem to infer the correct number of components of this mixture model. Our approach involves reframing the mixture model as a latent variable model, treating only the mixing coefficients as parameters, and subsequently approximating the marginal likelihood using an evidence lower bound framework. Our algorithm is evaluated on a large simulated dataset generated using a collection of different graph structures (band, hub, cluster, random, and scale-free).