  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
results on their convergence properties, e.g. much less than for gradient-based
sampling methods. In this work we analyse the behaviour of total variation
mixing times of Gibbs samplers targeting hierarchical models using tools from
Bayesian asymptotics. We obtain dimension-free convergence results 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.
