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The aim of reduced rank regression is to connect multiple response variablesto multiple predictors. This model is very popular, especially in biostatisticswhere multiple measurements on individuals can be re-used to predict multipleoutputs. Unfortunately, there are often missing data in such datasets, makingit difficult to use standard estimation tools. In this paper, we study theproblem of reduced rank regression where the response matrix is incomplete. Wepropose a quasi-Bayesian approach to this problem, in the sense that thelikelihood is replaced by a quasi-likelihood. We provide a tight oracleinequality, proving that our method is adaptive to the rank of the coefficientmatrix. We describe a Langevin Monte Carlo algorithm for the computation of theposterior mean. Numerical comparison on synthetic and real data show that ourmethod are competitive to the state-of-the-art where the rank is chosen bycross validation, and sometimes lead to an improvement.

Optimal quasi-Bayesian reduced rank regression with incomplete response