Variational inference for probabilistic poisson PCA

Many application domains, such as ecology or genomics, have to deal with multivariate non-Gaussian observations. A typical example is the joint observation of the respective abundances of a set of species in a series of sites aiming to understand the covariations between these species. The Gaussian setting provides a canonical way to model such dependencies but does not apply in general. We consider here the multivariate exponential family framework for which we introduce a generic model with multivariate Gaussian latent variables. We show that approximate maximum likelihood inference can be achieved via a variational algorithm for which gradient descent easily applies. We show that this setting enables us to account for covariates and offsets. We then focus on the case of the Poisson-lognormal model in the context of community ecology. We demonstrate the efficiency of our algorithm on microbial ecology datasets. We illustrate the importance of accounting for the effects of covariates to better understand interactions between species.