Coupling‐based convergence assessment of some Gibbs samplers for high‐dimensional Bayesian regression with shrinkage priors
Coupling‐based convergence assessment of some Gibbs samplers for high‐dimensional Bayesian regression with shrinkage priors
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DOI：
10.1111/rssb.12495
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发表时间：
2020-12
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影响因子：
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通讯作者：
N. Biswas;A. Bhattacharya;P. Jacob;J. Johndrow
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文献类型：
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作者：
N. Biswas;A. Bhattacharya;P. Jacob;J. Johndrow
研究了具有连续收缩先验的贝叶斯高维回归的马尔可夫链蒙特卡罗算法。这些算法的一个共同挑战是选择要执行的迭代次数。当每一次迭代都很昂贵时，这是至关重要的，就像处理现代数据集时一样，例如有数千行和数十万列的全基因组关联研究。我们开发了适合于具有收缩先验的高维回归设置的耦合技术，该技术能够实现实用的、非渐近的收敛诊断，而不依赖于轨迹图或长期渐近。通过为所考虑的算法建立几何漂移和最小化条件，我们证明了所提出的耦合具有有限的期望相遇时间。以一类包含“马蹄铁”的收缩先验为例，对所提出的联轴器的可扩展性进行了实证验证。我们发现的一个亮点是，在对100,000个协变量的回归中，不到1000次迭代足以让Gibbs采样器达到平稳。数值结果还说明了先验对耦合计算效率的影响，并建议在局部精度为半t分布且自由度大于1的情况下使用先验。
We consider Markov chain Monte Carlo (MCMC) algorithms for Bayesian high‐dimensional regression with continuous shrinkage priors. A common challenge with these algorithms is the choice of the number of iterations to perform. This is critical when each iteration is expensive, as is the case when dealing with modern data sets, such as genome‐wide association studies with thousands of rows and up to hundreds of thousands of columns. We develop coupling techniques tailored to the setting of high‐dimensional regression with shrinkage priors, which enable practical, non‐asymptotic diagnostics of convergence without relying on traceplots or long‐run asymptotics. By establishing geometric drift and minorization conditions for the algorithm under consideration, we prove that the proposed couplings have finite expected meeting time. Focusing on a class of shrinkage priors which includes the ‘Horseshoe’, we empirically demonstrate the scalability of the proposed couplings. A highlight of our findings is that less than 1000 iterations can be enough for a Gibbs sampler to reach stationarity in a regression on 100,000 covariates. The numerical results also illustrate the impact of the prior on the computational efficiency of the coupling, and suggest the use of priors where the local precisions are Half‐t distributed with degree of freedom larger than one.