  Estimating the diagonal entries of a matrix, that is not directly accessible
but only available as a linear operator in the form of a computer routine, is a
common necessity in many computational applications, especially in image
reconstruction and statistical inference. Here, methods of statistical
inference are used to improve the accuracy or the computational costs of matrix
probing methods to estimate matrix diagonals. In particular, the generalized
Wiener filter methodology, as developed within information field theory, is
shown to significantly improve estimates based on only a few sampling probes,
in cases in which some form of continuity of the solution can be assumed. The
strength, length scale, and precise functional form of the exploited
autocorrelation function of the matrix diagonal is determined from the probes
themselves. The developed algorithm is successfully applied to mock and real
world problems. These performance tests show that, in situations where a matrix
diagonal has to be calculated from only a small number of computationally
expensive probes, a speedup by a factor of 2 to 10 is possible with the
proposed method.
