  The cumulative distribution network (CDN) is a recently developed class of
probabilistic graphical models (PGMs) permitting a copula factorization, in
which the CDF, rather than the density, is factored. Despite there being much
recent interest within the machine learning community about copula
representations, there has been scarce research into the CDN, its amalgamation
with copula theory, and no evaluation of its performance. Algorithms for
inference, sampling, and learning in these models are underdeveloped compared
those of other PGMs, hindering widerspread use.
  One advantage of the CDN is that it allows the factors to be parameterized as
copulae, combining the benefits of graphical models with those of copula
theory. In brief, the use of a copula parameterization enables greater
modelling flexibility by separating representation of the marginals from the
dependence structure, permitting more efficient and robust learning. Another
advantage is that the CDN permits the representation of implicit latent
variables, whose parameterization and connectivity are not required to be
specified. Unfortunately, that the model can encode only latent relationships
between variables severely limits its utility.
  In this thesis, we present inference, learning, and sampling for CDNs, and
further the state-of-the-art. First, we explain the basics of copula theory and
the representation of copula CDNs. Then, we discuss inference in the models,
and develop the first sampling algorithm. We explain standard learning methods,
propose an algorithm for learning from data missing completely at random
(MCAR), and develop a novel algorithm for learning models of arbitrary
treewidth and size. Properties of the models and algorithms are investigated
through Monte Carlo simulations. We conclude with further discussion of the
advantages and limitations of CDNs, and suggest future work.
