Do higher-order moments improve inference of population dynamics?

Deciphering the interactions among host-associated microbial communities, such as the gut microbiome, is an important, but challenging task — there is constant flux of microbes and frequent environmental fluctuations driven by external factors and host immune response. At the same time, data collection is inherently constrained, as non-invasive sampling methods only capture snapshots of the system, and the number of available sampling points is often small. In such non-ideal cases, Bayesian parameter inference allows us not only to estimate model parameters from limited and noisy data, but also to quantify the uncertainty in those estimates. Studies using Bayesian techniques typically focus on fitting the mean dynamics to averaged data alone, neglecting the information provided by higher-order moments. In this study, we fit stochastic models of population dynamics to simulated microbiome data, and investigate whether the additional fitting of second-order moments, that is, fitting the variance and covariance observed across replicates to those expected from demographic stochasticity in the underlying model, can improve parameter inference by incorporating information that would otherwise be discarded. For simulated datasets, we observe a substantial improvement in parameter inference. However, we find that both the manner in which the first-order moments and second-order moments are combined in the distance function and the stopping criterion used during the optimization procedure substantially influence the quality of parameter inference.