  Bayesian multinomial logistic-normal (MLN) models are popular for the
analysis of sequence count data (e.g., microbiome or gene expression data) due
to their ability to model multivariate count data with complex covariance
structure. However, existing implementations of MLN models are limited to
handling small data sets due to the non-conjugacy of the multinomial and
logistic-normal distributions. We introduce MLN models which can be written as
marginally latent matrix-t process (LTP) models. Marginally LTP models describe
a flexible class of generalized linear regression, non-linear regression, and
time series models. We develop inference schemes for Marginally LTP models and,
through application to MLN models, demonstrate that our inference schemes are
both highly accurate and often 4-5 orders of magnitude faster than MCMC.
