  Beyond the linear regime, the power spectrum and higher order moments of the
matter field no longer capture all cosmological information encoded in density
fluctuations. While non-linear transforms have been proposed to extract this
information lost to traditional methods, up to now, the way to generalize these
techniques to discrete processes was unclear; ad hoc extensions had some
success. We pointed out in Carron and Szapudi (2013) that the logarithmic
transform approximates extremely well the optimal "sufficient statistics",
observables that extract all information from the (continuous) matter field.
Building on these results, we generalize optimal transforms to discrete galaxy
fields. We focus our calculations on the Poisson sampling of an underlying
lognormal density field. We solve and test the one-point case in detail, and
sketch out the sufficient observables for the multi-point case. Moreover, we
present an accurate approximation to the sufficient observables in terms of the
mean and spectrum of a non-linearly transformed field. We find that the
corresponding optimal non-linear transformation is directly related to the
maximum a posteriori Bayesian reconstruction of the underlying continuous field
with a lognormal prior as put forward in Kitaura et al (2010). Thus simple
recipes for realizing the sufficient observables can be built on previously
proposed algorithms that have been successfully implemented and tested in
simulations.
