In this talk I describe MAGIC, an efficient approach to covariance estimation and signal reconstruction for Gaussian random fields (MAGIC Allows Global Inference of Covariance). It solves a long-standing problem in the field of cosmic microwave background (CMB) data analysis but is in fact a general technique that can be applied to noisy, contaminated and incomplete or censored measurements of either spatial or temporal Gaussian random fields. In this talk I will phrase the method in a way that emphasizes its general structure and applicability but I comment on applications in the CMB context. The method allows the exploration of the full non-Gaussian joint posterior density of the signal and parameters in the covariance matrix (such as the power spectrum) given the data. It generalizes the familiar Wiener filter in that it automatically discovers signal correlations in the data as long as a noise model is specified and priors encode what is known about potential contaminants. The key methodological difference is that instead of attempting to evaluate the likelihood (or posterior density) or its derivatives, this method generates an asymptotically exact Monte Carlo sample from it. I present example applications to power spectrum estimation and signal reconstruction from measurements of the CMB. For these applications the method achieves speed-ups of many orders of magnitude compared to likelihood maximization techniques, while offering greater flexibility in modeling and a full characterization of the uncertainty in the estimates.
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