Gibbs samplers are popular algorithms to approximate posterior distributions arising from Bayesian hierarchical models. Despite their popularity and good empirical performances, however, there are still relatively few quantitative theoretical results on their scalability or lack thereof, e.g. much less than for gradient-based sampling methods. We introduce a novel technique to analyse the asymptotic behaviour of mixing times of Gibbs Samplers, based on tools of Bayesian asymptotics. We apply our methodology to high dimensional hierarchical models, obtaining dimension-free convergence results for Gibbs samplers under random data-generating assumptions, for a broad class of two-level models with generic likelihood function. Specific examples with Gaussian, binomial and categorical likelihoods are discussed.

翻译：吉布斯采样器是近似贝叶斯层次模型后验分布的流行算法。尽管这些算法应用广泛且经验表现良好，但其可扩展性（或缺乏可扩展性）的定量理论结果仍相对较少——例如远少于基于梯度的采样方法。我们提出了一种基于贝叶斯渐近性工具的新技术，用于分析吉布斯采样器混合时间的渐近行为。将该方法应用于高维层次模型，在随机数据生成假设下，针对一类具有通用似然函数的两层模型，获得了吉布斯采样器的维数无关收敛结果。具体讨论了高斯、二项分布和类别似然函数的实例。