In this thesis we develop a two component mixture model to perform a Bayesian regression. We implement our model computationally using the Gibbs sampler algorithm and apply it to a dataset of differences in time measurement between two clocks. The dataset has ``good" time measurements and ``bad" time measurements that were associated with the two components of our mixture model. From our theoretical work we show that latent variables are a useful tool to implement our Bayesian normal mixture model with two components. After applying our model to the data we found that the model reasonably assigned probabilities of occurrence to the two states of the phenomenon of study; it also identified two processes with the same slope, different intercepts and different variances. / McAnulty College and Graduate School of Liberal Arts; / Computational Mathematics / MS; / Thesis;

Thesis--Cornell. / Bibliography: p. 108-110.

In this thesis, a mixture-model cluster analysis technique under different covariance structures of the component densities is developed and presented, to capture the compactness, orientation, shape, and the volume of component clusters in one expert system to handle Gaussian high dimensional heterogeneous data sets to achieve flexibility in currently practiced cluster analysis techniques. Two approaches to parameter estimation are considered and compared; one using the Expectation-Maximization (EM) algorithm and another following a Bayesian framework using the Gibbs sampler. We develop and score several forms of the ICOMP criterion of Bozdogan (1994, 2004) as our fitness function; to choose the number of component clusters, to choose the correct component covariance matrix structure among nine candidate covariance structures, and to select the optimal parameters and the best fitting mixture-model. We demonstrate our approach on simulated datasets and a real large data set, focusing on early detection of breast cancer. We show that our approach improves the probability of classification error over the existing methods.

We consider a system of infinitely many hard balls in R^d undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with a local time term. We prove that the set of all equilibrium measures, solution of a detailed balance equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.

We consider an infinite system of hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite- dimensional Stochastic Differential Equation with an infinite-dimensional local time term. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also show that Gibbs measures are reversible measures.

We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional Stochastic Differential Equation with a local time term. We prove that the set of all equilibrium measures, solution of a Detailed Balance Equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.

The Ginibre gas is a Poisson point process dened on a space of loops related to the Feynman-Kac representation of the ideal Bose gas. Here we study thermodynamic limits of dierent ensembles via Martin-Dynkin boundary technique and show, in which way innitely long loops occur. This effect is the so-called Bose-Einstein condensation.

We prove in this paper an existence result for infinite-dimensional stationary interactive Brownian diffusions. The interaction is supposed to be small in the norm ||.||∞ but otherwise is very general, being possibly non-regular and non-Markovian. Our method consists in using the characterization of such diffusions as space-time Gibbs fields so that we construct them by space-time cluster expansions in the small coupling parameter.

We propose a new construction of point processes, which generalizes the class of infinitely divisible point processes. Examples are the quantum Boson and Fermion gases as well as the classical Gibbs point processes, where the interaction is given by a stable and regular pair potential.

In this thesis, we discuss some topics in random matrix theory which have applications to probability, statistics and quantum information theory. In Chapter 2, by relying on the spectral properties of an associated adjacency matrix, we find the distribution of the maximum of a Dyck path and show that it has the same distribution function as the unsigned Brownian excursion which was first derived in 1976 by Kennedy. We obtain a large and moderate deviation principle for the law of the maximum of a random Dyck path. Our result extends the results of Chung, Kennedy and Khorunzhiy and Marckert. In Chapter 3, we discuss a method of sampling called the Gibbs-slice sampler. This method is based on Neal's slice sampling combined with Gibbs sampling. In Chapter 4, we discuss several examples which have applications in physics and quantum information theory.