Uncertainty quantification for radio interferometric imaging: II. MAP estimation

Abstract

Uncertainty quantification is a critical missing component in radio\ninterferometric imaging that will only become increasingly important as the\nbig-data era of radio interferometry emerges. Statistical sampling approaches\nto perform Bayesian inference, like Markov Chain Monte Carlo (MCMC) sampling,\ncan in principle recover the full posterior distribution of the image, from\nwhich uncertainties can then be quantified. However, for massive data sizes,\nlike those anticipated from the Square Kilometre Array (SKA), it will be\ndifficult if not impossible to apply any MCMC technique due to its inherent\ncomputational cost. We formulate Bayesian inference problems with\nsparsity-promoting priors (motivated by compressive sensing), for which we\nrecover maximum a posteriori (MAP) point estimators of radio interferometric\nimages by convex optimisation. Exploiting recent developments in the theory of\nprobability concentration, we quantify uncertainties by post-processing the\nrecovered MAP estimate. Three strategies to quantify uncertainties are\ndeveloped: (i) highest posterior density credible regions; (ii) local credible\nintervals (cf. error bars) for individual pixels and superpixels; and (iii)\nhypothesis testing of image structure. These forms of uncertainty\nquantification provide rich information for analysing radio interferometric\nobservations in a statistically robust manner. Our MAP-based methods are\napproximately $10^5$ times faster computationally than state-of-the-art MCMC\nmethods and, in addition, support highly distributed and parallelised\nalgorithmic structures. For the first time, our MAP-based techniques provide a\nmeans of quantifying uncertainties for radio interferometric imaging for\nrealistic data volumes and practical use, and scale to the emerging big-data\nera of radio astronomy.\n