  We propose a novel approach, based on Principal Components Analysis, to the
use of Gamma-Ray Bursts (GRBs) as probes of cosmic star formation history (SFH)
up to very high redshifts. The main advantage of such approach is to avoid the
necessity of assuming an \textit{ad hoc} parameterization of the SFH. We first
validate the method by reconstructing a known SFH from Monte Carlo-generated
mock data. We then apply the method to the most recent \textit{Swift} data of
GRBs with known redshift and compare it against the SFH obtained by independent
methods. The main conclusion is that the level of star formation activity at $z
\approx 9.4$ could have been already as high as the present-day one ($\approx
0.01 M_\odot$ yr$^{-1}$ Mpc$^{-3}$). This is a factor 3-5 times higher than
deduced from high-$z$ galaxy searches through drop-out techniques. If true,
this might alleviate the long-standing problem of a photon-starving
reionization; it might also indicate that galaxies accounting for most of the
star formation activity at high redshift go undetected by even the most deep
searches.
