Probe combination in large galaxy surveys : Application of Fisher
  information and Shannon entropy to weak lensing
  This paper is aimed at developing a better understanding of the structure of
the information that in contained in galaxy surveys, so as to find optimal ways
to combine observables from such surveys. We first show how Jaynes' Maximal
Entropy Principle allows us, in the general case, to express the Fisher
information content of data sets in terms of the curvature of the Shannon
entropy surface with respect to the relevant observables. This allows us to
understand the Fisher information content of a data set, once a physical model
is specified, independently of the specific way that the data will be
processed, and without any assumptions of Gaussianity. This includes as a
special case the standard Fisher matrix prescriptions for Gaussian variables
widely used in the cosmological community, for instance for power spectra
extraction. As an application of this approach, we evaluate the prospects of a
joint analysis of weak lensing tracers up to second order in the shapes
distortions, in the case that the noise in each probe can be effectively
treated as model independent. These include the magnification, the two
ellipticity and the four flexion fields. At the two point level, we show that
the only effect of treating these observables in combination is a simple scale
dependent decrease of the noise contaminating the accessible spectrum of the
lensing E-mode. We provide simple bounds to its extraction by a combination of
such probes, as well as its quantitative evaluation when the correlations
between the noise variables for any two such probes can be neglected.
