The gut microbiota is a complex ecosystem composed mostly of bacteria interacting with each other and their environment. If the composition of the microbiota has proven to be a relevant biomarker for several diseases, the specific structure of gut microbiota has poorly been taken into account. Indeed, the microbial composition is described by discrete and sparse data, which also have a taxonomic structure. More precisely, each bacterium belongs to several groups that are hierarchically ordered from more precise (species) to less precise (domain). Although this taxonomic information is known, it remains unclear how it is related to the impact of a bacterium on its host. While some recent works (Chiquet et al. 2019) proposed a framework that accounts for the sparse and discrete nature of the microbiota, the impact of the taxonomic structure has not been investigated yet. In this work, we aim to introduce a new framework and dedicated algorithms that account for the sparse taxonomic abundance nature of the microbiota data. Our focus is to propose interpretable methods to unveil the complex interplay among bacterial species, with an application in the context of inflammatory bowel diseases.

The proposed approach, based on Poisson Log-Normal models, accounts for Markov dependencies to include the taxonomic tree structure linking the bacteria while enabling taxa of different branches to influence each other. This not only extends the scope of interaction modeling but also aligns with the inherent complexity and diversity of microbial communities. Additionally, we present a novel variational approach that incorporates the Markov structure of the posterior distribution to enhance the precision of the variational estimation. Such variational families allow us to obtain theoretical guarantees on latent state estimation and data reconstruction.

The generative structure of the model enables us to challenge our proposed framework on artificial data. The performance is assessed using typical microbial metrics like alpha and beta diversity as well as standard statistical indicators like graph dissimilarities. Our analysis explores the model characteristics, highlighting its capacity to faithfully replicate underlying microbial dynamics within the gut microbiome. To assess real-world applications, we conduct an empirical investigation on a cohort of patients diagnosed with Crohn's disease. The model is systematically benchmarked against state-of-the-art algorithms, providing a comprehensive analysis of its efficiency in capturing complex microbial interactions associated with pathological conditions.