  We present a non-parametric Lagrangian biasing model and fit the ratio of the
halo and mass densities at the field level using the mass-weighted halo field
in the AbacusSummit simulations at $z=0.5$. Unlike the perturbative halo bias
model widely used in interpreting the observed large-scale structure traced by
galaxies, we find a non-negative halo-to-mass ratio that increases
monotonically with the linear overdensity $\delta_1$ in the initial Lagrangian
space. The bias expansion, however, does not guarantee non-negativity of the
halo counts, and may give rising halo number counts at negative overdensities.
The shape of the halo-to-mass ratio is unlikely to be described by a polynomial
of $\delta_1$ and other quantities. Especially for massive halos with
$6\times10^{12}\ h^{-1}\ M_\odot$, the halo-to-mass ratio starts soaring up at
$\delta_1>0$, substantially different from the predictions of the bias
expansion. We show that for the halo masses we consider ($M>3\times10^{11}\
h^{-1}\ M_\odot$) a non-parametric halo-to-mass ratio as a function of
$\delta_1$ and its local derivative $\nabla^2\delta_1$ can recover the halo
power spectra to sub-percent level for wavenumbers $k=0.01-0.1\ h\ {\rm
Mpc}^{-1}$ given a proper smoothing scale to filter the initial density field,
even though we do not fit the power spectrum directly. However, there is mild
dependence of the recovery of the halo power spectrum on the smoothing scale
and other input parameters. At $k<0.01\ h\ {\rm Mpc}^{-1}$ and for massive
halos with $M>6\times10^{12}\ h^{-1}\ M_\odot$, our non-parametric model leads
to a few percent overestimation of the halo power spectrum, indicating the need
for larger or multiple smoothing scales. The halo-to-mass ratios obtained
qualitatively agree with intuitions from extended Press-Schechter theory. We
compare our framework to the bias expansion and discuss possible extensions.
