Convergence Analysis of MCMC Algorithms for Bayesian Multivariate Linear Regression with Non‐Gaussian Errors
研究了当误差为非高斯分布时，用于贝叶斯多元线性回归的数据增强和Haar PX-DA算法的马尔可夫链几何遍历性条件，给出了混合密度需满足的要求。
Abstract When Gaussian errors are inappropriate in a multivariate linear regression setting, it is often assumed that the errors are iid from a distribution that is a scale mixture of multivariate normals. Combining this robust regression model with a default prior on the unknown parameters results in a highly intractable posterior density. Fortunately, there is a simple data augmentation (DA) algorithm and a corresponding Haar PX‐DA algorithm that can be used to explore this posterior. This paper provides conditions (on the mixing density) for geometric ergodicity of the Markov chains underlying these Markov chain Monte Carlo algorithms. Letting d denote the dimension of the response, the main result shows that the DA and Haar PX‐DA Markov chains are geometrically ergodic whenever the mixing density is generalized inverse Gaussian, log‐normal, inverted Gamma (with shape parameter larger than d /2) or Fréchet (with shape parameter larger than d /2). The results also apply to certain subsets of the Gamma, F and Weibull families.