  The neutral hydrogen (HI) content of dark matter haloes forms an intermediate
state in the baryon cycle that connects the hot shock-heated gas and cold
star-forming gas in haloes. Measurement of the relationship between HI mass and
halo mass therefore puts important constraints on galaxy formation models. We
combine radio observations of HI in emission at low redshift ($z\sim 0$) with
optical/UV observations of HI in absorption at high redshift ($1<z<4$) to
derive constraints on the evolution of the HI-mass halo-mass (HIHM) relation
from redshift $z=4$ to $z=0$. We find that one can model the HIHM relation
similar to the stellar-mass halo-mass (SHM) relation at $z \sim 0$. At $z=0$,
haloes with mass $10^{11.7}$ M$_\odot$ have the highest HI mass fraction ($\sim
1\%$), which is about four times smaller than their stellar mass fraction. We
model the evolution of the HIHM relation in a manner similar to that of the SHM
relation. Combining this parameterisation with a redshift- and mass-dependent
modified Navarro-Frenk-White (NFW) profile for the HI density within a halo, we
draw constraints on the evolution of the HIHM relation from the observed HI
column density, incidence rate, and clustering bias at high redshift. We
compare these findings with results from hydrodynamical simulations and other
approaches in the literature, and find the models to be consistent with each
other at the $68\%$ confidence level.
