  Distributions over rankings are used to model data in a multitude of real
world settings such as preference analysis and political elections. Modeling
such distributions presents several computational challenges, however, due to
the factorial size of the set of rankings over an item set. Some of these
challenges are quite familiar to the artificial intelligence community, such as
how to compactly represent a distribution over a combinatorially large space,
and how to efficiently perform probabilistic inference with these
representations. With respect to ranking, however, there is the additional
challenge of what we refer to as human task complexity users are rarely willing
to provide a full ranking over a long list of candidates, instead often
preferring to provide partial ranking information. Simultaneously addressing
all of these challenges i.e., designing a compactly representable model which
is amenable to efficient inference and can be learned using partial ranking
data is a difficult task, but is necessary if we would like to scale to
problems with nontrivial size. In this paper, we show that the recently
proposed riffled independence assumptions cleanly and efficiently address each
of the above challenges. In particular, we establish a tight mathematical
connection between the concepts of riffled independence and of partial
rankings. This correspondence not only allows us to then develop efficient and
exact algorithms for performing inference tasks using riffled independence
based represen- tations with partial rankings, but somewhat surprisingly, also
shows that efficient inference is not possible for riffle independent models
(in a certain sense) with observations which do not take the form of partial
rankings. Finally, using our inference algorithm, we introduce the first method
for learning riffled independence based models from partially ranked data.
