Reconstruction of signals with unknown spectra in information field
  theory with parameter uncertainty
  The optimal reconstruction of cosmic metric perturbations and other signals
requires knowledge of their power spectra and other parameters. If these are
not known a priori, they have to be measured simultaneously from the same data
used for the signal reconstruction. We formulate the general problem of signal
inference in the presence of unknown parameters within the framework of
information field theory. We develop a generic parameter uncertainty
renormalized estimation (PURE) technique and address the problem of
reconstructing Gaussian signals with unknown power-spectrum with five different
approaches: (i) separate maximum-a-posteriori power spectrum measurement and
subsequent reconstruction, (ii) maximum-a-posteriori power reconstruction with
marginalized power-spectrum, (iii) maximizing the joint posterior of signal and
spectrum, (iv) guessing the spectrum from the variance in the Wiener filter
map, and (v) renormalization flow analysis of the field theoretical problem
providing the PURE filter. In all cases, the reconstruction can be described or
approximated as Wiener filter operations with assumed signal spectra derived
from the data according to the same recipe, but with differing coefficients.
All of these filters, except the renormalized one, exhibit a perception
threshold in case of a Jeffreys prior for the unknown spectrum. Data modes,
with variance below this threshold do not affect the signal reconstruction at
all. Filter (iv) seems to be similar to the so called Karhune-Loeve and
Feldman-Kaiser-Peacock estimators for galaxy power spectra used in cosmology,
which therefore should also exhibit a marginal perception threshold if
correctly implemented. We present statistical performance tests and show that
the PURE filter is superior to the others.
