  The aim of reduced rank regression is to connect multiple response variables
to multiple predictors. This model is very popular, especially in biostatistics
where multiple measurements on individuals can be re-used to predict multiple
outputs. Unfortunately, there are often missing data in such datasets, making
it difficult to use standard estimation tools. In this paper, we study the
problem of reduced rank regression where the response matrix is incomplete. We
propose a quasi-Bayesian approach to this problem, in the sense that the
likelihood is replaced by a quasi-likelihood. We provide a tight oracle
inequality, proving that our method is adaptive to the rank of the coefficient
matrix. We describe a Langevin Monte Carlo algorithm for the computation of the
posterior mean. Numerical comparison on synthetic and real data show that our
method are competitive to the state-of-the-art where the rank is chosen by
cross validation, and sometimes lead to an improvement.
