  We present the first unbiased determination of spin-dependent, or polarized,
Parton Distribution Functions (PDFs) of the proton. A statistically sound
representation of the corresponding uncertainties is achieved by means of the
NNPDF methodology: this was formerly developed for unpolarized distributions
and is now generalized to the polarized here for the first time. The features
of the procedure, based on robust statistical tools (Monte Carlo sampling for
error propagation, neural networks for PDF parametrization, genetic algorithm
for their minimization, and possibly reweighting for including new data samples
without refitting), are illustrated in detail. Different sets of polarized PDFs
are obtained at next-to-leading order accuracy in perturbative quantum
chromodynamics, based on both fixed-target inclusive deeply-inelastic
scattering data and the most recent polarized collider data. A quantitative
appraisal on the potential role of future measurements at an Electron-Ion
Collider is also presented. We study the stability of our results upon the
variation of several theoretical and methodological assumptions and we present
a detailed investigation of the first moments of our polarized PDFs, compared
to other recent analyses. We find that the uncertainty on the gluon
distribution from available data was substantially underestimated in previous
determinations; in particular, we emphasize that a large contribution to the
gluon may arise from the unmeasured small-x region, against the common belief
that this is actually rather small. We demonstrate that an Electron-Ion
Collider would provide evidence for a possible large gluon contribution to the
nucleon spin, though with a sizable residual uncertainty.
