Browsing by Subject "Dirichlet process"

With the development of modern data collection approaches, researchers may collect hundreds to millions of variables, yet may not need to utilize all explanatory variables available in predictive models. Hence, choosing models that consist of a subset of variables often becomes a crucial step. In linear regression, variable selection not only reduces model complexity, but also prevents over-fitting. From a Bayesian perspective, prior specification of model parameters plays an important role in model selection as well as parameter estimation, and often prevents over-fitting through shrinkage and model averaging.

We develop two novel hierarchical priors for selection and model averaging, for Generalized Linear Models (GLMs) and normal linear regression, respectively. They can be considered as "spike-and-slab" prior distributions or more appropriately "spike- and-bell" distributions. Under these priors we achieve dimension reduction, since their point masses at zero allow predictors to be excluded with positive posterior probability. In addition, these hierarchical priors have heavy tails to provide robust- ness when MLE's are far from zero.

Zellner's g-prior is widely used in linear models. It preserves correlation structure among predictors in its prior covariance, and yields closed-form marginal likelihoods which leads to huge computational savings by avoiding sampling in the parameter space. Mixtures of g-priors avoid fixing g in advance, and can resolve consistency problems that arise with fixed g. For GLMs, we show that the mixture of g-priors using a Compound Confluent Hypergeometric distribution unifies existing choices in the literature and maintains their good properties such as tractable (approximate) marginal likelihoods and asymptotic consistency for model selection and parameter estimation under specific values of the hyper parameters.

While the g-prior is invariant under rotation within a model, a potential problem with the g-prior is that it inherits the instability of ordinary least squares (OLS) estimates when predictors are highly correlated. We build a hierarchical prior based on scale mixtures of independent normals, which incorporates invariance under rotations within models like ridge regression and the g-prior, but has heavy tails like the Zeller-Siow Cauchy prior. We find this method out-performs the gold standard mixture of g-priors and other methods in the case of highly correlated predictors in Gaussian linear models. We incorporate a non-parametric structure, the Dirichlet Process (DP) as a hyper prior, to allow more flexibility and adaptivity to the data.

Two-sample comparison is a fundamental problem in statistics. Given two samples of data, the interest lies in understanding whether the two samples were generated by the same distribution or not. Traditional two-sample comparison methods are not suitable for modern data where the underlying distributions are multivariate and highly multi-modal, and the differences across the distributions are often locally concentrated. The focus of this thesis is to develop novel statistical methodology for two-sample comparison which is effective in such scenarios. Tools from the nonparametric Bayesian literature are used to flexibly describe the distributions. Additionally, the two-sample comparison problem is decomposed into a collection of local tests on individual parameters describing the distributions. This strategy not only yields high statistical power, but also allows one to identify the nature of the distributional difference. In many real-world applications, detecting the nature of the difference is as important as the existence of the difference itself. Generalizations to multi-sample comparison and more complex statistical problems, such as multi-way analysis of variance, are also discussed.

I consider the problem of clustering multiple related groups of data. My approach entails mixture models in the context of hierarchical Dirichlet processes, focusing on their ability to perform inference on the unknown number of components in the mixture, as well as to facilitate the sharing of information and borrowing of strength across the various data groups. Here, I build upon the hierarchical Dirichlet process model proposed by Muller et al. (2004), revising some relevant aspects of the model, as well as improving the MCMC sampler's convergence by combining local Gibbs sampler moves with global Metropolis-Hastings split-merge moves. I demonstrate the strengths of my model by employing it to cluster both synthetic and real datasets.

The immune system is constantly faced with the daunting task of protecting the host from a large number of ever-evolving pathogens. In vertebrates, the immune response results from the interplay of two cellular systems: the innate immunity and the adaptive immunity. In the past decades, dendritic cells have emerged as major players in the modulation of the immune response, being one of the primary links between these two branches of the immune system.

Dendritic cells are pathogen-sensing cells that alert the rest of the immune system of the presence of infection. The signals sent by dendritic cells result in the recruitment of the appropriate cell types and molecules required for effectively clearing the infection. A question of utmost importance in our understanding of the immune response and our ability to manipulate it in the development of vaccines and therapies is: "How do dendritic cells translate the various cues they perceive from the environment into different signals that specifically activate the appropriate parts of the immune system that result in an immune response streamlined to clear the given pathogen?"

Here we have developed computational and statistical methods aimed to address specific aspects of this question. In particular, understanding how dendritic cells ultimately modulate the immune response requires an understanding of the subtleties of their maturation process in response to different environmental signals. Hence, the first part of this dissertation focuses on elucidating the changes in the transcriptional

program of dendritic cells in response to the detection of two common pathogen- associated molecules, LPS and CpG. We have developed a method based on Langevin and Dirichlet processes to model and cluster gene expression temporal data, and have used it to identify, on a large scale, genes that present unique and common transcriptional behaviors in response to these two stimuli. Additionally, we have also investigated a different, but related, aspect of dendritic cell modulation of the adaptive immune response. In the second part of this dissertation, we present a method to predict peptides that will bind to MHC molecules, a requirement for the activation of pathogen-specific T cells. Together, these studies contribute to the elucidation of important aspects of dendritic cell biology.

Bayesian nonparametric methods are useful for modeling data without having to define the complexity of the entire model a priori, but rather allowing for this complexity to be determined by the data. Two problems considered in this dissertation are the number of components in a mixture model, and the number of factors in a latent factor model, for which the Dirichlet process and the beta process are the two respective Bayesian nonparametric priors selected for handling these issues.

The flexibility of Bayesian nonparametric priors arises from the prior's definition over an infinite dimensional parameter space. Therefore, there are theoretically an infinite number of latent components and an infinite number of latent factors. Nevertheless, draws from each respective prior will produce only a small number of components or factors that appear in a given data set. As mentioned, the number of these components and factors, and their corresponding parameter values, are left for the data to decide.

This dissertation is split between novel practical applications and novel theoretical results for these priors. For the Dirichlet process, we investigate stick-breaking representations for the finite Dirichlet process and their application to novel sampling techniques, as well as a novel mixture modeling framework that incorporates multiple modalities within a data set. For the beta process, we present a new stick-breaking construction for the infinite-dimensional prior, and consider applications to image interpolation problems and dictionary learning for compressive sensing.

In most machine learning approaches, it is usually assumed that data are complete. When data are partially missing due to various reasons, for example, the failure of a subset of sensors, image corruption or inadequate medical measurements, many learning methods designed for complete data cannot be directly applied. In this dissertation we treat two kinds of problems with incomplete data using non-parametric Bayesian approaches: classification with incomplete features and analysis of low-rank matrices with missing entries.

Incomplete data in classification problems are handled by assuming input features to be generated from a mixture-of-experts model, with each individual expert (classifier) defined by a local Gaussian in feature space. With a linear classifier associated with each Gaussian component, nonlinear classification boundaries are achievable without the introduction of kernels. Within the proposed model, the number of components is theoretically ``infinite'' as defined by a Dirichlet process construction, with the actual number of mixture components (experts) needed inferred based upon the data under test. With a higher-level DP we further extend the classifier for analysis of multiple related tasks (multi-task learning), where model components may be shared across tasks. Available data could be augmented by this way of information transfer even when tasks are only similar in some local regions of feature space, which is particularly critical for cases with scarce incomplete training samples from each task. The proposed algorithms are implemented using efficient variational Bayesian inference and robust performance is demonstrated on synthetic data, benchmark data sets, and real data with natural missing values.

Another scenario of interest is to complete a data matrix with entries missing. The recovery of missing matrix entries is not possible without additional assumptions on the matrix under test, and here we employ the common assumption that the matrix is low-rank. Unlike methods with a preset fixed rank, we propose a non-parametric Bayesian alternative based on the singular value decomposition (SVD), where missing entries are handled naturally, and the number of underlying factors is imposed to be small and inferred in the light of observed entries. Although we assume missing at random, the proposed model is generalized to incorporate auxiliary information including missingness features. We also make a first attempt in the matrix-completion community to acquire new entries actively. By introducing a probit link function, we are able to handle counting matrices with the decomposed low-rank matrices latent. The basic model and its extensions are validated on

synthetic data, a movie-rating benchmark and a new data set presented for the first time.

We propose nonparametric Bayesian models for supervised dimension

reduction and regression problems. Supervised dimension reduction is

a setting where one needs to reduce the dimensionality of the

predictors or find the dimension reduction subspace and lose little

or no predictive information. Our first method retrieves the

dimension reduction subspace in the inverse regression framework by

utilizing a dependent Dirichlet process that allows for natural

clustering for the data in terms of both the response and predictor

variables. Our second method is based on ideas from the gradient

learning framework and retrieves the dimension reduction subspace

through coherent nonparametric Bayesian kernel models. We also

discuss and provide a new rationalization of kernel regression based

on nonparametric Bayesian models allowing for direct and formal

inference on the uncertain regression functions. Our proposed models

apply for high dimensional cases where the number of variables far

exceed the sample size, and hold for both the classical setting of

Euclidean subspaces and the Riemannian setting where the marginal

distribution is concentrated on a manifold. Our Bayesian perspective

adds appropriate probabilistic and statistical frameworks that allow

for rich inference such as uncertainty estimation which is important

for measuring the estimates. Formal probabilistic models with

likelihoods and priors are given and efficient posterior sampling

can be obtained by Markov chain Monte Carlo methodologies,

particularly Gibbs sampling schemes. For the supervised dimension

reduction as the posterior draws are linear subspaces which are

points on a Grassmann manifold, we do the posterior inference with

respect to geodesics on the Grassmannian. The utility of our

approaches is illustrated on simulated and real examples.

Bayesian nonparametric models, such as the Gaussian process and the Dirichlet process, have been extensively applied for target kinematics modeling in various applications including environmental monitoring, traffic planning, endangered species tracking, dynamic scene analysis, autonomous robot navigation, and human motion modeling. As shown by these successful applications, Bayesian nonparametric models are able to adjust their complexities adaptively from data as necessary, and are resistant to overfitting or underfitting. However, most existing works assume that the sensor measurements used to learn the Bayesian nonparametric target kinematics models are obtained a priori or that the target kinematics can be measured by the sensor at any given time throughout the task. Little work has been done for controlling the sensor with bounded field of view to obtain measurements of mobile targets that are most informative for reducing the uncertainty of the Bayesian nonparametric models. To present the systematic sensor planning approach to leaning Bayesian nonparametric models, the Gaussian process target kinematics model is introduced at first, which is capable of describing time-invariant spatial phenomena, such as ocean currents, temperature distributions and wind velocity fields. The Dirichlet process-Gaussian process target kinematics model is subsequently discussed for modeling mixture of mobile targets, such as pedestrian motion patterns.

Novel information theoretic functions are developed for these introduced Bayesian nonparametric target kinematics models to represent the expected utility of measurements as a function of sensor control inputs and random environmental variables. A Gaussian process expected Kullback Leibler divergence is developed as the expectation of the KL divergence between the current (prior) and posterior Gaussian process target kinematics models with respect to the future measurements. Then, this approach is extended to develop a new information value function that can be used to estimate target kinematics described by a Dirichlet process-Gaussian process mixture model. A theorem is proposed that shows the novel information theoretic functions are bounded. Based on this theorem, efficient estimators of the new information theoretic functions are designed, which are proved to be unbiased with the variance of the resultant approximation error decreasing linearly as the number of samples increases. Computational complexities for optimizing the novel information theoretic functions under sensor dynamics constraints are studied, and are proved to be NP-hard. A cumulative lower bound is then proposed to reduce the computational complexity to polynomial time.

Three sensor planning algorithms are developed according to the assumptions on the target kinematics and the sensor dynamics. For problems where the control space of the sensor is discrete, a greedy algorithm is proposed. The efficiency of the greedy algorithm is demonstrated by a numerical experiment with data of ocean currents obtained by moored buoys. A sweep line algorithm is developed for applications where the sensor control space is continuous and unconstrained. Synthetic simulations as well as physical experiments with ground robots and a surveillance camera are conducted to evaluate the performance of the sweep line algorithm. Moreover, a lexicographic algorithm is designed based on the cumulative lower bound of the novel information theoretic functions, for the scenario where the sensor dynamics are constrained. Numerical experiments with real data collected from indoor pedestrians by a commercial pan-tilt camera are performed to examine the lexicographic algorithm. Results from both the numerical simulations and the physical experiments show that the three sensor planning algorithms proposed in this dissertation based on the novel information theoretic functions are superior at learning the target kinematics with

little or no prior knowledge