  Microbes are everywhere, including in and on our bodies, and have been shown
to play key roles in a variety of prevalent human diseases. Consequently, there
has been intense interest in the design of bacteriotherapies or "bugs as
drugs," which are communities of bacteria administered to patients for specific
therapeutic applications. Central to the design of such therapeutics is an
understanding of the causal microbial interaction network and the population
dynamics of the organisms. In this work we present a Bayesian nonparametric
model and associated efficient inference algorithm that addresses the key
conceptual and practical challenges of learning microbial dynamics from time
series microbe abundance data. These challenges include high-dimensional (300+
strains of bacteria in the gut) but temporally sparse and non-uniformly sampled
data; high measurement noise; and, nonlinear and physically non-negative
dynamics. Our contributions include a new type of dynamical systems model for
microbial dynamics based on what we term interaction modules, or learned
clusters of latent variables with redundant interaction structure (reducing the
expected number of interaction coefficients from $O(n^2)$ to $O((\log n)^2)$);
a fully Bayesian formulation of the stochastic dynamical systems model that
propagates measurement and latent state uncertainty throughout the model; and
introduction of a temporally varying auxiliary variable technique to enable
efficient inference by relaxing the hard non-negativity constraint on states.
We apply our method to simulated and real data, and demonstrate the utility of
our technique for system identification from limited data and gaining new
biological insights into bacteriotherapy design.
