  Sampling-based algorithms are classical approaches to perform Bayesian
inference in inverse problems. They provide estimators with the associated
credibility intervals to quantify the uncertainty on the estimators. Although
these methods hardly scale to high dimensional problems, they have recently
been paired with optimization techniques, such as proximal and splitting
approaches, to address this issue. Such approaches pave the way to distributed
samplers, splitting computations to make inference more scalable and faster. We
introduce a distributed Split Gibbs sampler (SGS) to efficiently solve such
problems involving distributions with multiple smooth and non-smooth functions
composed with linear operators. The proposed approach leverages a recent
approximate augmentation technique reminiscent of primal-dual optimization
methods. It is further combined with a block-coordinate approach to split the
primal and dual variables into blocks, leading to a distributed
block-coordinate SGS. The resulting algorithm exploits the hypergraph structure
of the involved linear operators to efficiently distribute the variables over
multiple workers under controlled communication costs. It accommodates several
distributed architectures, such as the Single Program Multiple Data and
client-server architectures. Experiments on a large image deblurring problem
show the performance of the proposed approach to produce high quality estimates
with credibility intervals in a small amount of time. Supplementary material to
reproduce the experiments is available online.
