On properties of fractional posterior in generalized reduced-rank regression
研究了广义线性模型在降秩回归框架下的分数后验性质，放宽了链接函数需为典则链接的限制，证明了后验的一致性和集中性，且无需事先知道参数矩阵的秩，对模型误设也具有鲁棒性。
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. We also investigate the impact of model misspecification, demonstrating the robustness of our approach in such cases. Numerical simulations and real data analyses are conducted to illustrate the promising performance of our approach compared to the state-of-the-art method.