inverting inverse covariance matrices

On my flight across the Atlantic I took photos from my seat which I today assembled into this video:

I also finished reading and making comments on Berry Holl's (Lund) PhD thesis. He has worked out expansions for inverting enormous Gaia-like inverse covariance matrices—inverse covariance matrices tend to be simple and sparse, covariance matrices tend to be complex and dense—and he has shown that Gaia can deliver on its promises despite the expected radiation damage to its CCDs. This radiation damage leads to charge transfer inefficiency, which leads to changes in the point-spread function in the scan (charge-transfer) direction on the CCDs. This leads to timing residuals which in principle affect astrometric measurements. However, the multiple scan angles at which Gaia hits each field saves it, even if the CTI is evolving with time and doesn't match exactly any of the (somewhat heuristic) models.

Holl impressed me 1.5 years ago and it will be an honor to play the (formal) role of opponent at his defense.

One thing I need to work out (for my own good) is how inverting the inverse covariance matrix relates to marginalization. The diagonal elements of the inverse covariance matrix are like the inverse uncertainty variances holding all other parameters fixed, whereas the diagonal elements of the covariance matrix are like the uncertainties marginalizing out all other parameters. That's all cool. But inverting the inverse covariance matrix is something any responsible frequentist must do; marginalization is only permitted for Bayesians. Do you see why I am confused? I am not confused about the math; I am confused about the meaning.