  This article describes a robust algorithm to estimate a conditional
probability density f(t|x) as a non-parametric smooth regression function. It
is based on a neural network and the Bayesian interpretation of the network
output as a posteriori probabability. The network is trained using example
events from history or simulation, which define the underlying probability
density f(t,x).
  Once trained, the network is applied on new, unknown examples x, for which it
can predict the probability distribution of the target variable t.
Event-by-event knowledge of the smooth function f(t|x) can be very useful, e.g.
in maximum likelihood fits or for forecasting tasks. No assumptions are
necessary about the distribution, and non-Gaussian tails are accounted for
automatically. Important quantities like median, mean value, left and right
standard deviations, moments and expectation values of any function of t are
readily derived from it.
  The algorithm can be considered as an event-by-event unfolding and leads to
statistically optimal reconstruction. The largest benefit of the method lies in
complicated problems, when the measurements x are only relatively weakly
correlated to the output t. As to assure optimal generalisation features and to
avoid overfitting, the networks are regularised by extended versions of weight
decay. The regularisation parameters are determined during the online-learning
of the network by relations obtained from Bayesian statistics.
  Some toy Monte Carlo tests and first real application examples from
high-energy physics and econometry are discussed.
