We utilize the ensemble of trees framework, a tractable mixture over super-exponential number of tree-structured distributions, to develop a new model for multivariate density estimation. The model is based on the construction of tree-structured copulas -- multivariate distributions with uniform on [0,1] marginals. By averaging over all possible tree structures, the new model can approximate distributions with complex variable dependencies. We propose an EM algorithm to estimate the parameters for these tree-averaged models for both the real-valued and the categorical case. Based on the tree-averaged framework, we propose a new model for joint precipitation amounts data on networks of rain stations.

Statistical models on full and partial rankings of n items are often of limited practical use for large n due to computational consideration. We explore the use of non-parametric models for partially ranked data and derive efficient procedures for their use for large n. The derivations are largely possible through combinatorial and algebraic manipulations based on the lattice of partial rankings. In particular, we demonstrate for the first time a non-parametric coherent and consistent model capable of efficiently aggregating partially ranked data of different types.

Permutations are ubiquitous in many real world problems, such as voting, rankings and data association. Representing uncertainty over permutations is challenging, since there are $n!$ possibilities, and typical compact representations, such as graphical models, cannot capture the mutual exclusivity constraints associated with permutations. In this paper, we use the ``low-frequency'' terms of a Fourier decomposition to represent such distributions compactly. We present \emph{Kronecker conditioning}, a new general, efficient approach for maintaining these distributions directly in the Fourier domain. Low order Fourier-based approximations can lead to functions that do not correspond to valid distributions. To address this problem, we present an efficient quadratic program defined directly in the Fourier domain to project the approximation onto the polytope of legal marginal distributions. We demonstrate the effectiveness of our approach on a real camera-based multi-people tracking setting.

In order to represent state in controlled, partially observable, stochastic dynamical systems, some sort of sufficient statistic for history is necessary. Predictive representations of state (PSRs) capture state as statistics of the future. We introduce a new model of such systems called the ``Exponential family PSR,'' which defines as state the time-varying parameters of an exponential family distribution which models n sequential observations in the future. This choice of state representation explicitly connects PSRs to state-of-the-art probabilistic modeling, which allows us to take advantage of current efforts in high-dimensional density estimation, and in particular, graphical models and maximum entropy models. We present a parameter learning algorithm based on maximum likelihood, and we show how a variety of current approximate inference methods apply. We evaluate the quality of our model with reinforcement learning by directly evaluating the control performance of the model.