  Performing exact Bayesian inference for complex models is computationally
intractable. Markov chain Monte Carlo (MCMC) algorithms can provide reliable
approximations of the posterior distribution but are expensive for large
datasets and high-dimensional models. A standard approach to mitigate this
complexity consists in using subsampling techniques or distributing the data
across a cluster. However, these approaches are typically unreliable in
high-dimensional scenarios. We focus here on a recent alternative class of MCMC
schemes exploiting a splitting strategy akin to the one used by the celebrated
alternating direction of multipliers (ADMM) optimization algorithm. These
methods appear to provide empirically state-of-the-art performance but their
theoretical behavior in high dimension is currently unknown. In this paper, we
propose a detailed theoretical study of one of these algorithms known as the
split Gibbs sampler. Under regularity conditions, we establish explicit
convergence rates for this scheme using Ricci curvature and coupling ideas. We
support our theory with numerical illustrations.
