We present a probabilistic model of events in continuous time in which each
event triggers a Poisson process of successor events. The ensemble of observed
events is thereby modeled as a superposition of Poisson processes. Efficient
inference is feasible under this model with an EM algorithm. Moreover, the EM
algorithm can be implemented as a distributed algorithm, permitting the model
to be applied to very large datasets. We apply these techniques to the modeling
of Twitter messages and the revision history of Wikipedia.

In this work, we establish novel connections between the Bayesian
nonparametric clustering and featural paradigms by considering the problem of
admixture modeling. We examine the Dirichlet process-and its unnormalized
Poisson point process generation via the gamma process-on the traditional
clustering side of Bayesian nonparametrics. On the featural side, we examine
the beta process and introduce a new model, the beta negative binomial process
(BNBP), for admixture modeling.

This work introduces SubMF, a parallel divide-and-conquer framework for noisy
matrix factorization. SubMF divides a large-scale matrix factorization task
into smaller subproblems, solves each subproblem in parallel using an arbitrary
base matrix factorization algorithm, and combines the subproblem solutions
using techniques from randomized matrix approximation. Our experiments with
collaborative filtering, video background modeling, and simulated data
demonstrate the near-linear to super-linear speed-ups attainable with this
approach.

Inspired by Random Forests (RF) in the context of classification, we propose
a new clustering ensemble method---Cluster Forests (CF). Geometrically, CF
randomly probes a high-dimensional data cloud to obtain "good local
clusterings" and then aggregates via spectral clustering to obtain cluster
assignments for the whole dataset. The search for good local clusterings is
guided by a cluster quality measure $\kappa$. CF progressively improves each
local clustering in a fashion that resembles the tree growth in RF.

Statistics is a uniquely difficult field to convey to the uninitiated. It
sits astride the abstract and the concrete, the theoretical and the applied. It
has a mathematical flavor and yet it is not simply a branch of mathematics. Its
core problems blend into those of the disciplines that probe into the nature of
intelligence and thought, in particular philosophy, psychology and artificial
intelligence. Debates over foundational issues have waxed and waned, but the
field has not yet arrived at a single foundational perspective.

Heavy-tailed distributions are frequently used to enhance the robustness of
regression and classification methods to outliers in output space. Often,
however, we are confronted with ``outliers'' in input space, which are isolated
observations in sparsely populated regions. We show that heavy-tailed
stochastic processes (which we construct from Gaussian processes via a copula),
can be used to improve robustness of regression and classification estimators
to such outliers by selectively shrinking them more strongly in sparse regions
than in dense regions.

Many complex dynamical phenomena can be effectively modeled by a system that
switches among a set of conditionally linear dynamical modes. We consider two
such models: the switching linear dynamical system (SLDS) and the switching
vector autoregressive (VAR) process. Our Bayesian nonparametric approach
utilizes a hierarchical Dirichlet process prior to learn an unknown number of
persistent, smooth dynamical modes.

We present the nested Chinese restaurant process (nCRP), a stochastic process
which assigns probability distributions to infinitely-deep,
infinitely-branching trees. We show how this stochastic process can be used as
a prior distribution in a Bayesian nonparametric model of document collections.
Specifically, we present an application to information retrieval in which
documents are modeled as paths down a random tree, and the preferential
attachment dynamics of the nCRP leads to clustering of documents according to
sharing of topics at multiple levels of abstraction.