Consistent estimation of the spectrum of trace class data augmentation algorithms

Markov chain Monte Carlo is widely used in a variety of scientific applications to generate approximate samples from intractable distributions. A thorough understanding of the convergence and mixing properties of these Markov chains can be obtained by studying the spectrum of the associated Markov operator. While several methods to bound/estimate the second largest eigenvalue are available in the literature, very few general techniques for consistent estimation of the entire spectrum have been proposed. Existing methods for this purpose require the Markov transition density to be available in closed form, which is often not true in practice, especially in modern statistical applications. In this paper, we propose a novel method to consistently estimate the entire spectrum of a general class of Markov chains arising from a popular and widely used statistical approach known as Data Augmentation. The transition densities of these Markov chains can often only be expressed as intractable integrals. We illustrate the applicability of our method using real and simulated data.Comment: 43 pages (including Appendix), 3 figures; final versio

Convergence properties of Gibbs samplers for Bayesian probit regression with proper priors

The Bayesian probit regression model (Albert and Chib (1993)) is popular and widely used for binary regression. While the improper flat prior for the regression coefficients is an appropriate choice in the absence of any prior information, a proper normal prior is desirable when prior information is available or in modern high dimensional settings where the number of coefficients ($p$) is greater than the sample size ($n$). For both choices of priors, the resulting posterior density is intractable and a Data Dugmentation (DA) Markov chain is used to generate approximate samples from the posterior distribution. Establishing geometric ergodicity for this DA Markov chain is important as it provides theoretical guarantees for constructing standard errors for Markov chain based estimates of posterior quantities. In this paper, we first show that in case of proper normal priors, the DA Markov chain is geometrically ergodic *for all* choices of the design matrix $X$, $n$ and $p$ (unlike the improper prior case, where $n \geq p$ and another condition on $X$ are required for posterior propriety itself). We also derive sufficient conditions under which the DA Markov chain is trace-class, i.e., the eigenvalues of the corresponding operator are summable. In particular, this allows us to conclude that the Haar PX-DA sandwich algorithm (obtained by inserting an inexpensive extra step in between the two steps of the DA algorithm) is strictly better than the DA algorithm in an appropriate sense.Comment: 40 pages, 6 figures; typos correcte

Surface Functionalized Gold Nanoparticle Applications in Catalysis and Lipid-Nanoparticle Assemblies

The general objectives of my dissertation are to devise strategies to impart unique functionality to gold nanoparticles (AuNPs) through surface modification and utilizing such surface-functionalized AuNP systems to 1) facilitate colloidal catalyst recovery and reuse while maintaining activity and 2) measure the interactions of engineered AuNPs with biological membranes as related to stimuli-responsive liposomal therapeutics. AuNP surface functionalization is an essential step in the synthesis, stability and functionality-specific applications of AuNPs. Due to the ability to control AuNP shape, size and surface properties, AuNP applications have seen an exponential growth in various sectors that include catalysis, biomedicine, drug-delivery, sensing, imaging, energy applications, etc. Solution-based synthesis of surface functionalized AuNPs through the Turkevich and Brust synthesis methods produce reproducible narrow size-distribution colloidal AuNPs and provides a reliable platform to investigate effects of surface functionality on various applications. Chapters 1 and 2 introduces the concept of AuNP as catalyst and experimentally compares the two major categories of catalyst; colloidal versus supported catalyst). Chapters 3-4 shows the successful development of a platform of colloidal AuNPs which are recovered from an aqueous medium through pH manipulation and illustrates how ligand structure on the AuNP surface is a primary factor determining catalytic activity, stability and recoverability. Chapter 5 is aimed at investigating interactions of engineered AuNP with the lipid membranes of small unilamellar vesicles (SUV) and depicts substantial membrane softening from AuNP inclusions. The colloidal systems were investigated through various characterization tools such as calorimetry, light scattering, spectroscopy and advanced electron microscopy. Additionally, this work has employed elastic and inelastic neutron scattering techniques heavily as a result of extensive collaboration with government user facilities such as National Institute of Standards and Technology and Oak Ridge National Laboratory. Overall, in my doctoral research, I have successfully manipulated AuNP properties through surface functionalization and assessed AuNP applications in two major areas: (1) sustainable catalysis and (2) lipid-nanoparticle assemblies as a platform to quantify membrane biomechanical properties. The current work sets the groundwork for expansion of such functionalized nanoparticle systems (including other metal NPs such as Pt, Pd, Ag, etc.) to a much larger application base in catalysis, nano-bio interactions, nanomedicine, drug delivery, and sensors to name a few

Uterine artery Doppler study for the prediction and the severity of the hypertensive disorders during pregnancy

Background: Hypertensive disorder in pregnancy is one of the major cause of maternal and perinatal mortality and morbidity worldwide particularly in developing countries. In developed countries, maternal mortality rate varies from 4-40 per 1 lakh live birth. In developing countries, it varies from 100-700 with India having 178 per 1 lakh live birth. Objectives were to study of uterine artery Doppler for the prediction of hypertensive disorders and its severity, Sensitivity and Specificity of Uterine Artery Doppler and their comparison for the prediction and the severity of hypertensive disorders during pregnancy.Methods: This is a prospective study. Approximately hundred patients with hypertensive disorders during pregnancy attending the OPD and IPD in Obstetrics and Gynecology Department will be included. All the patients in this study group will be subjected to Ultrasonographic evaluation of the pregnancy along with the Arterial Doppler of both the Uterine Artery and Umbilical Artery will be done.Results: Using uterine artery Doppler Study in combination is significantly useful in early prediction of PIH having specificity and sensitivity of Uterine artery Doppler study – Pi Index as 91.67% and 85.71 %, Ri Index as 87.5% and 71.43% and diastolic notch as 94.44% and 92.85 % respectively. The use of uterine artery Doppler Study as an important tool for early prediction of PIH and has a lot of prognostic value.Conclusions: The predictive accuracy of uterine artery Doppler study using Pi Index and Ri Index is better in the detection of early-onset PIH, than the late-onset disease. Thereby, it will help in the long run to prevent the increasing maternal and fetal morbidity and mortality

Modification of Liposome Release Characteristics with Metal Nanoparticles

Gold nanoparticles have been getting increasing amount of attention due to their stability, optical properties and low toxicity in the field of consumer products and biomedicines. Gold nanoparticles with various surface chemistries have been synthesized in order to take advantage of the size, shape and morphology dependent properties of the nanoparticles which vary greatly from the bulk properties. These properties of gold nanoparticles have made them attractive in optical sensing, targeted drug delivery, bio imaging, chemical, catalytic and photo thermal applications. These surface functionalized nanoparticles were used to study nanoparticle-lipid bilayer interactions using lipid bilayer permeability studies, where permeability of small unilamellar vesicles (SUVs) were studied spectrofluorometrically by measuring the efflux of encapsulated fluorescent marker dye from the vesicles in the presence of the surface functionalized gold and silver nanoparticles. SUVs containing highly self-quenched entrapped fluorescein was prepared from common lipids and subjected to nanoparticles. Initially the fluorescence is low due to self-quenching but increases dramatically as the entrapped dye leaks out of the liposomes (with time, temperature or due to interactions with the nanoparticles resulting in the change of stability or permeability of the lipid bilayer). This extent of change in fluorescence signal in the form of relative fluorescence of the leaked marker dye was used to determine the change in membrane permeability