We introduce a novel class of Bayesian mixtures for normal linear regression models which incorporates a further Gaussian random component for the distribution of the predictor variables. The proposed cluster-weighted model aims to encompass potential heterogeneity in the distribution of the response variable as well as in the multivariate distribution of the covariates for detecting signals relevant to the underlying latent structure. Of particular interest are potential signals originating from: (i) the linear predictor structures of the regression models and (ii) the covariance structures of the covariates. We model these two components using a lasso shrinkage prior for the regression coefficients and a graphical-lasso shrinkage prior for the covariance matrices. A fully Bayesian approach is followed for estimating the number of clusters, by treating the number of mixture components as random and implementing a trans-dimensional telescoping sampler. Alternative Bayesian approaches based on overfitting mixture models or using information criteria to select the number of components are also considered. The proposed method is compared against EM type implementation, mixtures of regressions and mixtures of experts. The method is illustrated using a set of simulation studies and a biomedical dataset.

翻译：我们提出了一类新颖的贝叶斯正态线性回归模型混合方法，该方法为预测变量的分布引入了一个额外的高斯随机分量。所提出的聚类加权模型旨在同时捕捉响应变量分布中的潜在异质性，以及协变量多元分布中的异质性，以检测与底层潜在结构相关的信号。特别感兴趣的潜在信号来源于：（i）回归模型的线性预测结构，以及（ii）协变量的协方差结构。我们使用回归系数的lasso压缩先验和协方差矩阵的图lasso压缩先验来建模这两个分量。通过将混合分量数量视为随机变量，并实现跨维望远镜采样器，我们采用完全贝叶斯方法来估计簇的数量。同时考虑了基于过拟合混合模型或使用信息准则选择分量数量的替代贝叶斯方法。所提出的方法与EM类实现、回归混合及专家混合进行了比较。通过一系列模拟研究和一个生物医学数据集对该方法进行了验证。