On properties of fractional posterior in generalized reduced-rank regression

Reduced rank regression (RRR) is a widely employed model for investigating the linear association between multiple response variables and a set of predictors. While RRR has been extensively explored in various works, the focus has predominantly been on continuous response variables, overlooking other types of outcomes. This study shifts its attention to the Bayesian perspective of generalized linear models (GLM) within the RRR framework. In this work, we relax the requirement for the link function of the generalized linear model to be canonical. We examine the properties of fractional posteriors in GLM within the RRR context, where a fractional power of the likelihood is utilized. By employing a spectral scaled Student prior distribution, we establish consistency and concentration results for the fractional posterior. Our results highlight adaptability, as they do not necessitate prior knowledge of the rank of the parameter matrix. These results are in line with those found in frequentist literature. Additionally, an examination of model mis-specification is undertaken, underscoring the effectiveness of our approach in such scenarios.