  We propose a method to infer causal structures containing both discrete and
continuous variables. The idea is to select causal hypotheses for which the
conditional density of every variable, given its causes, becomes smooth. We
define a family of smooth densities and conditional densities by second order
exponential models, i.e., by maximizing conditional entropy subject to first
and second statistical moments. If some of the variables take only values in
proper subsets of R^n, these conditionals can induce different families of
joint distributions even for Markov-equivalent graphs.
  We consider the case of one binary and one real-valued variable where the
method can distinguish between cause and effect. Using this example, we
describe that sometimes a causal hypothesis must be rejected because
P(effect|cause) and P(cause) share algorithmic information (which is untypical
if they are chosen independently). This way, our method is in the same spirit
as faithfulness-based causal inference because it also rejects non-generic
mutual adjustments among DAG-parameters.
