  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 co-variations 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.
