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.

翻译：降秩回归是一种广泛用于研究多个响应变量与一组预测变量之间线性关联的模型。尽管降秩回归已在多项工作中得到深入探讨，但关注点主要集中在连续响应变量上，忽略了其他类型的结果变量。本研究将关注点转向降秩回归框架内广义线性模型的贝叶斯视角。本文放宽了广义线性模型链接函数需为典则函数的要求，研究了降秩回归背景下广义线性模型中分数后验的性质——即利用似然函数的分数幂。通过采用谱缩放的Student先验分布，我们建立了分数后验的一致性和集中性结果。这些结果具有适应性，因为它们不要求事先知道参数矩阵的秩。这些结论与频率学派文献中的发现一致。此外，本文还进行了模型误设的分析，进一步验证了该方法在误设场景下的有效性。