  Understanding $\textit{galaxy bias}$ -- that is the statistical relation
between matter and galaxies -- is of key importance for extracting cosmological
information from galaxy surveys. While the bias function $f$ -- that is the
probability of forming galaxy in a region with a given density field -- is
usually approximated through a parametric expansion, we show here, that it can
also be measured directly from simulations in a non-parameteric way. Our
measurements show that the Lagrangian bias function is very close to a Gaussian
for halo selections of any mass. Therefore, we newly introduce a Gaussian bias
model with several intriguing properties: (1) It predicts only strictly
positive probabilities $f > 0$ (unlike expansion models), (2) It has a simple
analytic renormalized form and (3) It behaves gracefully in many scenarios
where the classical expansion converges poorly. We show that the Gaussian bias
model describes the galaxy environment distribution $p(\delta | \mathrm{g})$,
the scale dependent bias function $f$ and the renormalized bias function $F$ of
haloes and galaxies generally equally well or significantly better than a
second order expansion with the same number of parameters. We suggest that a
Gaussian bias approach may enhance the range of validity of bias schemes where
the canonical expansion converges poorly and further, that it may make new
applications possible, since it guarantees the positivity of predicted galaxy
densities.
