  The relationship between galaxies and haloes is central to the description of
galaxy formation, and a fundamental step towards extracting precise
cosmological information from galaxy maps. However, this connection involves
several complex processes that are interconnected. Machine Learning methods are
flexible tools that can learn complex correlations between a large number of
features, but are traditionally designed as deterministic estimators. In this
work, we use the IllustrisTNG300-1 simulation and apply neural networks in a
binning classification scheme to predict probability distributions of central
galaxy properties, namely stellar mass, colour, specific star formation rate,
and radius, using as input features the halo mass, concentration, spin, age,
and the overdensity on a scale of 3 $h^{-1}$ Mpc. The model captures the
intrinsic scatter in the relation between halo and galaxy properties, and can
thus be used to quantify the uncertainties related to the stochasticity of the
galaxy properties with respect to the halo properties. In particular, with our
proposed method, one can define and accurately reproduce the properties of the
different galaxy populations in great detail. We demonstrate the power of this
tool by directly comparing traditional single-point estimators and the
predicted joint probability distributions, and also by computing the power
spectrum of a large number of tracers defined on the basis of the predicted
colour-stellar mass diagram. We show that the neural networks reproduce
clustering statistics of the individual galaxy populations with excellent
precision and accuracy.
