Stochastic models allow improved inference of microbiome interactions from time series data.

How can we figure out how the different microbes interact within microbiomes? To combine theoretical models and experimental data, we often fit a deterministic model for the mean dynamics of a system to averaged data. However, in the averaging procedure a lot of information from the data is lost-and a deterministic model may be a poor representation of a stochastic reality. Here, we develop an inference method for microbiomes based on the idea that both the experiment and the model are stochastic. Starting from a stochastic model, we derive dynamical equations not only for the average, but also for higher statistical moments of the microbial abundances. We use these equations to infer distributions of the interaction parameters that best describe the biological experimental data-improving identifiability and precision. The inferred distributions allow us to make predictions but also to distinguish between fairly certain parameters and those for which the available experimental data does not give sufficient information. Compared to related approaches, we derive expressions that also work for the relative abundance of microbes, enabling us to use conventional metagenome data, and account for cases where not a single host, but only replicate hosts, can be tracked over time.