  We develop information field theory (IFT) as a means of Bayesian inference on
spatially distributed signals, the information fields. A didactical approach is
attempted. Starting from general considerations on the nature of measurements,
signals, noise, and their relation to a physical reality, we derive the
information Hamiltonian, the source field, propagator, and interaction terms.
Free IFT reproduces the well known Wiener-filter theory. Interacting IFT can be
diagrammatically expanded, for which we provide the Feynman rules in position-,
Fourier-, and spherical harmonics space, and the Boltzmann-Shannon information
measure. The theory should be applicable in many fields. However, here, two
cosmological signal recovery problems are discussed in their IFT-formulation.
1) Reconstruction of the cosmic large-scale structure matter distribution from
discrete galaxy counts in incomplete galaxy surveys within a simple model of
galaxy formation. We show that a Gaussian signal, which should resemble the
initial density perturbations of the Universe, observed with a strongly
non-linear, incomplete and Poissonian-noise affected response, as the processes
of structure and galaxy formation and observations provide, can be
reconstructed thanks to the virtue of a response-renormalization flow equation.
2) We design a filter to detect local non-linearities in the cosmic microwave
background, which are predicted from some Early-Universe inflationary
scenarios, and expected due to measurement imperfections. This filter is the
optimal Bayes' estimator up to linear order in the non-linearity parameter and
can be used even to construct sky maps of non-linearities in the data.
