We present a new statistical framework called hidden Markov Dirichlet process (HMDP) to jointly model the genetic recombinations among possibly infinite number of founders and the coalescence-with-mutation events in the resulting genealogies. TheHMDP posits that a haplotype of genetic markers is generated by a sequence of recombination events that select an ancestor for each locus from an unbounded set of founders according to a 1st-order Markov transition process. Conjoining this process with a mutation model, our method accommodates both between-lineage recombination and within-lineage sequence variations, and leads to a compact and natural interpretation of the population structure and inheritance process underlying haplotype data. We have developed an efficient sampling algorithm forHMDP based on a two-level nested Pólya urn scheme. On both simulated and real SNP haplotype data, our method performs competitively or significantly better than extant methods in uncovering the recombination hotspots along chromosomal loci;and in addition it also infers the ancestral genetic patterns and offers a highly accurate map of ancestral compositions of modern populations.

Data sets that characterize human activity over time through collections of timestamped eventsor counts are of increasing interest in application areas as humancomputer interaction,video surveillance, and Web data analysis. We propose a nonparametric Bayesian framework for modeling collections of such data. In particular, we use a Dirichlet process framework for learning a set of intensity functions corresponding to different categories, which form a basis set for representing individualtime-periods (e.g., several days) depending on which categories the time-periods are assigned to. This allows the model to learn in a data-driven fashion what "factors" are generating the observations on a particular day, including (forexample) weekday versus weekend effects or day-specific effects corresponding tounique (single-day) occurrences of unusual behavior, sharing information where appropriate to obtain improved estimates of the behavior associated with each category. Applications to real-world data sets of count data involving both vehicles and people are used to illustrate the technique.

Perceptual Bistability refers to the phenomenon of spontaneously switching between twoor more interpretations of an image under continuous viewing. Although switchingbehavior is increasingly well characterized, the origins remain elusive. We propose that perceptual switching naturally arises from the brain's search for best interpretations while performing Bayesian inference. In particular, we propose that the brain explores a posterior distribution over image interpretations ata rapid time scale via a sampling-like process and updates its interpretation when a sampled interpretation is better than the discounted value of its current interpretation. Weformalize the theory, explicitly derive switching rate distributions and discuss qualitative properties of the theory including the effect of changes in the posterior distribution on switching rates. Finally, predictions of the theory are shown to be consistent with measured changes in human switching dynamics to Necker cube stimuli induced by context.

Propagation based methods are developed and rigorously assessed.

Scoring structures of undirected graphical models by means of evaluating the marginal likelihood is very hard. The main reason is the presence of the partition functionwhich is intractable to evaluate, let alone integrate over. We propose to approximate the marginal likelihood by employing two levels of approximation: we assume normality of the posterior (the Laplace approximation) and approximate allremaining intractable quantities using belief propagation and the linear response approximation.

We focus on the problem of estimating the graph structure associated with a discrete Markov random field.

Their machine counterparts have historically been unable to compete with this ability, until now.

The ill-posed nature of the MEG/EEG source localization problem requires the incorporation of prior assumptions when choosing an appropriate solution out of an infinite set of candidates. Bayesian methods are useful in this capacity because they allow these assumptions to be explicitly quantified. Recently, a number of empirical Bayesian approaches have been proposed that attempt a form of model selection by using the data to guide the search for an appropriate prior. While seemingly quite different in many respects, we apply a unifying framework based on automatic relevance determination (ARD) that elucidates various attributes of these methods and suggests directions for improvement. We also derive theoretical propertiesof this methodology related to convergence, local minima, and localization bias and explore connections with established algorithms.

We introduce binary matrix factorization, a novel model for unsupervised matrix decomposition.The decomposition is learned by fitting a nonparametric Bayesian probabilistic model with binary latent variables to a matrix of dyadic data. Unlike bi-clustering models, which assign each row or column to a single cluster based on a categorical hidden feature, our binary feature model reflects the prior belief that items and attributes can be associated with more than one latent cluster at a time. We provide simple learning and inference rules for this new model and show how to extend it to an infinite model in which the number of features is not a priori fixed but is allowed to grow with the size of the data.

Many problems that arise in machine learning domain deal with nonlinearity and quite often demand users to obtain global optimal solutions rather than local optimal ones. Optimization problems are inherent in machine learning algorithms and hence many methods in machine learning were inherited from the optimization literature. Popularly known as the initialization problem, the ideal set of parameters required will significantly depend on the given initialization values. The recently developed TRUST-TECH (TRansformation Under STability-reTaining Equilibria CHaracterization) methodology systematically explores the subspace of the parameters to obtain a complete set of local optimal solutions. In this thesis work, we propose TRUST-TECH based methods for solving several optimization and machine learning problems. Two stages namely, the local stage and the neighborhood-search stage, are repeated alternatively in the solution space to achieve improvements in the quality of the solutions. Our methods were tested on both synthetic and real datasets and the advantages of using this novel framework are clearly manifested. This framework not only reduces the sensitivity to initialization, but also allows the flexibility for the practitioners to use various global and local methods that work well for a particular problem of interest. Other hierarchical stochastic algorithms like evolutionary algorithms and smoothing algorithms are also studied and frameworks for combining these methods with TRUST-TECH have been proposed and evaluated on several test systems.