  Uncertainty quantification is a critical missing component in radio
interferometric imaging that will only become increasingly important as the
big-data era of radio interferometry emerges. Since radio interferometric
imaging requires solving a high-dimensional, ill-posed inverse problem,
uncertainty quantification is difficult but also critical to the accurate
scientific interpretation of radio observations. Statistical sampling
approaches to perform Bayesian inference, like Markov Chain Monte Carlo (MCMC)
sampling, can in principle recover the full posterior distribution of the
image, from which uncertainties can then be quantified. However, traditional
high-dimensional sampling methods are generally limited to smooth (e.g.
Gaussian) priors and cannot be used with sparsity-promoting priors. Sparse
priors, motivated by the theory of compressive sensing, have been shown to be
highly effective for radio interferometric imaging. In this article proximal
MCMC methods are developed for radio interferometric imaging, leveraging
proximal calculus to support non-differential priors, such as sparse priors, in
a Bayesian framework. Furthermore, three strategies to quantify uncertainties
using the recovered posterior distribution are developed: (i) local
(pixel-wise) credible intervals to provide error bars for each individual
pixel; (ii) highest posterior density credible regions; and (iii) hypothesis
testing of image structure. These forms of uncertainty quantification provide
rich information for analysing radio interferometric observations in a
statistically robust manner.
