  We propose a new method for parameter learning in Bayesian networks with
qualitative influences. This method extends our previous work from networks of
binary variables to networks of discrete variables with ordered values. The
specified qualitative influences correspond to certain order restrictions on
the parameters in the network. These parameters may therefore be estimated
using constrained maximum likelihood estimation. We propose an alternative
method, based on the isotonic regression. The constrained maximum likelihood
estimates are fairly complicated to compute, whereas computation of the
isotonic regression estimates only requires the repeated application of the
Pool Adjacent Violators algorithm for linear orders. Therefore, the isotonic
regression estimator is to be preferred from the viewpoint of computational
complexity. Through experiments on simulated and real data, we show that the
new learning method is competitive in performance to the constrained maximum
likelihood estimator, and that both estimators improve on the standard
estimator.
