Title: Uncertainty quantification for radio interferometric imaging: II. MAP estimation,
Abstract: 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. 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, for massive data sizes,
like those anticipated from the Square Kilometre Array (SKA), it will be
difficult if not impossible to apply any MCMC technique due to its inherent
computational cost. We formulate Bayesian inference problems with
sparsity-promoting priors (motivated by compressive sensing), for which we
recover maximum a posteriori (MAP) point estimators of radio interferometric
images by convex optimisation. Exploiting recent developments in the theory of
probability concentration, we quantify uncertainties by post-processing the
recovered MAP estimate. Three strategies to quantify uncertainties are
developed: (i) highest posterior density credible regions; (ii) local credible
intervals (cf. error bars) for individual pixels and superpixels; 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. Our MAP-based methods are
approximately $10^5$ times faster computationally than state-of-the-art MCMC
methods and, in addition, support highly distributed and parallelised
algorithmic structures. For the first time, our MAP-based techniques provide a
means of quantifying uncertainties for radio interferometric imaging for
realistic data volumes and practical use, and scale to the emerging big-data
era of radio astronomy.