Geometric ergodicity for Bayesian shrinkage models
Geometric ergodicity for Bayesian shrinkage models
复制标题
贝叶斯收缩模型的几何遍历性
DOI：
10.1214/14-ejs896
复制
发表时间：
2014
影响因子：
1.1
通讯作者：
K. Khare
中科院分区：
文献类型：
--
作者：
Subhadip Pal;K. Khare
近年来，已经开发了大量的各种连续收缩率，用于高维设置中的标准线性回归模型的贝叶斯分析。我们考虑两个这样的先验，狄利克雷-拉普拉斯先验（在Bhattacharya等人（2013）中开发的）和正态伽马先验（在（Gran和Brown，2010）中开发的）。对于Dirichlet-Laplace和Normal-Gamma先验，Gibbs抽样马尔可夫链已被开发用于从相应的后验分布生成近似样本。我们表明，通过使用漂移和minorization为基础的分析，吉布斯抽样马尔可夫链对应于上述模型是几何遍历。建立这些马尔可夫链的几何遍历性是至关重要的，因为它为使用马尔可夫链CLT提供了理论依据，然后可以用于获得基于马尔可夫链的后验量估计的渐近标准误差。本文中的两个Gibbs采样器都使用广义逆高斯（GIG）分布作为条件分布之一。我们的收敛性分析的一个新的贡献是使用漂移函数，其中包括正常随机变量的负分数幂项，以解决GIG分布的存在。
: In recent years, a large variety of continuous shrinkage pri- ors have been developed for a Bayesian analysis of the standard linear regression model in high dimensional settings. We consider two such pri- ors, the Dirichlet-Laplace prior (developed in Bhattacharya et al. (2013)), and the Normal-Gamma prior (developed in (Griffin and Brown, 2010)). For both Dirichlet-Laplace and Normal-Gamma priors, Gibbs sampling Markov chains have been developed to generate approximate samples from the cor- responding posterior distributions. We show by using a drift and minorization based analysis that the Gibbs sampling Markov chains corresponding to the aforementioned models are geometrically ergodic. Establishing geometric ergodicity of these Markov chains is crucial, as it provides theoretical justification for the use of Markov chain CLT, which can then be used to obtain asymptotic standard errors for Markov chain based estimates of posterior quantities. Both Gibbs samplers in the paper use the Generalized Inverse Gaussian (GIG) distribution, as one of the conditional distribu- tions. A novel contribution of our convergence analysis is the use of drift functions which include terms that are negative fractional powers of normal random variables, to tackle the presence of the GIG distribution.