  The ability to obtain reliable point estimates of model parameters is of
crucial importance in many fields of physics. This is often a difficult task
given that the observed data can have a very high number of dimensions. In
order to address this problem, we propose a novel approach to construct
parameter estimators with a quantifiable bias using an order expansion of
highly compressed deep summary statistics of the observed data. These summary
statistics are learned automatically using an information maximising loss.
Given an observation, we further show how one can use the constructed
estimators to obtain approximate Bayes computation (ABC) posterior estimates
and their corresponding uncertainties that can be used for parameter inference
using Gaussian process regression even if the likelihood is not tractable. We
validate our method with an application to the problem of cosmological
parameter inference of weak lensing mass maps. We show in that case that the
constructed estimators are unbiased and have an almost optimal variance, while
the posterior distribution obtained with the Gaussian process regression is
close to the true posterior and performs better or equally well than comparable
methods.
