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Microbiome data generated by next generation sequencing continue to flourish. There is a need for statistical models that can decode microbes' reaction to the environment and interactions among microbes simultaneously. The model should have the ability to correctly incorporate prior knowledge from controlled experiments that are oftentimes conditioned on other responses. We introduce a novel Bayesian conditional auto-regressive (CAR) LASSO model to infer a sparse network structure with nodes for responses and for predictors and whose edges all represent conditional dependence, not conditional among responses and marginal between responses and predictors. We also propose an adaptive extension of the CAR LASSO model so that different shrinkage can be applied to different edges which allows the incorporation of edge-specific prior knowledge. Indeed, the conditional representation of our model coefficients and adaptivity allow us to adequately encode prior knowledge obtained by specific experimental interventions and agrees with the experimenter's intuition on average behavior of nodes under experiments. In addition, our model is able to equally handle small and big data and is computationally inexpensive through an efficient Gibbs sampling algorithm. With hierarchical structure, we extend the model to binary, counting and compositional responses by adding an appropriate sampling distribution to the core Normal model. Finally, we apply our model to two real-life microbial composition datasets: one related to human gut and one related to soil.

We hereby developed a Markov random field based spatial-explicit community occupancy model that accounts for spatial auto-correlation and interspecific interactions in occupancy simultaneously while also account for interspecific interaction in detection. Simulation results showed the new model can distinguish different mechanisms. We applied this new model to camera trap data of Fisher(Pekania pennanti)-Marten(Martes americana) and Coyote(Canis latrans)-Fox(Vulpes vulpes) system in Apostle Island National Lakeshore. Results showed the observed partition pattern between marten and fisher may be better explained by a flipped mainland-island source-sink pattern rather than competition, while we detected some evidence that on top of the mainland-island source-sink pattern, there was a positive association between fox and coyote than deserve further study.

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