  We extend the log-mean linear parameterization introduced by Roverato et al.
(2013) for binary data to discrete variables with arbitrary number of levels,
and show that also in this case it can be used to parameterize bi-directed
graph models. Furthermore, we show that the log-mean linear parameterization
allows one to simultaneously represent marginal independencies among variables
and marginal independencies that only appear when certain levels are collapsed
into a single one. We illustrate the application of this property by means of
an example based on genetic association studies involving single-nucleotide
polymorphisms. More generally, this feature provides a natural way to reduce
the parameter count, while preserving the independence structure, by means of
substantive constraints that give additional insight into the association
structure of the variables.
