Bayesian Inference in Large-scale Problems

Many modern applications fall into the category of "large-scale" statistical problems, in which both the number of observations n and the number of features or parameters p may be large. Many existing methods focus on point estimation, despite the continued relevance of uncertainty quantification in the sciences, where the number of parameters to estimate often exceeds the sample size, despite huge increases in the value of n typically seen in many fields. Thus, the tendency in some areas of industry to dispense with traditional statistical analysis on the basis that "n=all" is of little relevance outside of certain narrow applications. The main result of the Big Data revolution in most fields has instead been to make computation much harder without reducing the importance of uncertainty quantification. Bayesian methods excel at uncertainty quantification, but often scale poorly relative to alternatives. This conflict between the statistical advantages of Bayesian procedures and their substantial computational disadvantages is perhaps the greatest challenge facing modern Bayesian statistics, and is the primary motivation for the work presented here.

Two general strategies for scaling Bayesian inference are considered. The first is the development of methods that lend themselves to faster computation, and the second is design and characterization of computational algorithms that scale better in n or p. In the first instance, the focus is on joint inference outside of the standard problem of multivariate continuous data that has been a major focus of previous theoretical work in this area. In the second area, we pursue strategies for improving the speed of Markov chain Monte Carlo algorithms, and characterizing their performance in large-scale settings. Throughout, the focus is on rigorous theoretical evaluation combined with empirical demonstrations of performance and concordance with the theory.

One topic we consider is modeling the joint distribution of multivariate categorical data, often summarized in a contingency table. Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. In Chapter 2, we derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions.

Latent class models for the joint distribution of multivariate categorical, such as the PARAFAC decomposition, data play an important role in the analysis of population structure. In this context, the number of latent classes is interpreted as the number of genetically distinct subpopulations of an organism, an important factor in the analysis of evolutionary processes and conservation status. Existing methods focus on point estimates of the number of subpopulations, and lack robust uncertainty quantification. Moreover, whether the number of latent classes in these models is even an identified parameter is an open question. In Chapter 3, we show that when the model is properly specified, the correct number of subpopulations can be recovered almost surely. We then propose an alternative method for estimating the number of latent subpopulations that provides good quantification of uncertainty, and provide a simple procedure for verifying that the proposed method is consistent for the number of subpopulations. The performance of the model in estimating the number of subpopulations and other common population structure inference problems is assessed in simulations and a real data application.

In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis--Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. In Chapter 4 we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis--Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback-Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even in relatively small samples. The proposed approximation provides a computationally scalable and principled approach to regularized estimation and approximate Bayesian inference for log-linear models.

Another challenging and somewhat non-standard joint modeling problem is inference on tail dependence in stochastic processes. In applications where extreme dependence is of interest, data are almost always time-indexed. Existing methods for inference and modeling in this setting often cluster extreme events or choose window sizes with the goal of preserving temporal information. In Chapter 5, we propose an alternative paradigm for inference on tail dependence in stochastic processes with arbitrary temporal dependence structure in the extremes, based on the idea that the information on strength of tail dependence and the temporal structure in this dependence are both encoded in waiting times between exceedances of high thresholds. We construct a class of time-indexed stochastic processes with tail dependence obtained by endowing the support points in de Haan's spectral representation of max-stable processes with velocities and lifetimes. We extend Smith's model to these max-stable velocity processes and obtain the distribution of waiting times between extreme events at multiple locations. Motivated by this result, a new definition of tail dependence is proposed that is a function of the distribution of waiting times between threshold exceedances, and an inferential framework is constructed for estimating the strength of extremal dependence and quantifying uncertainty in this paradigm. The method is applied to climatological, financial, and electrophysiology data.

The remainder of this thesis focuses on posterior computation by Markov chain Monte Carlo. The Markov Chain Monte Carlo method is the dominant paradigm for posterior computation in Bayesian analysis. It has long been common to control computation time by making approximations to the Markov transition kernel. Comparatively little attention has been paid to convergence and estimation error in these approximating Markov Chains. In Chapter 6, we propose a framework for assessing when to use approximations in MCMC algorithms, and how much error in the transition kernel should be tolerated to obtain optimal estimation performance with respect to a specified loss function and computational budget. The results require only ergodicity of the exact kernel and control of the kernel approximation accuracy. The theoretical framework is applied to approximations based on random subsets of data, low-rank approximations of Gaussian processes, and a novel approximating Markov chain for discrete mixture models.

Data augmentation Gibbs samplers are arguably the most popular class of algorithm for approximately sampling from the posterior distribution for the parameters of generalized linear models. The truncated Normal and Polya-Gamma data augmentation samplers are standard examples for probit and logit links, respectively. Motivated by an important problem in quantitative advertising, in Chapter 7 we consider the application of these algorithms to modeling rare events. We show that when the sample size is large but the observed number of successes is small, these data augmentation samplers mix very slowly, with a spectral gap that converges to zero at a rate at least proportional to the reciprocal of the square root of the sample size up to a log factor. In simulation studies, moderate sample sizes result in high autocorrelations and small effective sample sizes. Similar empirical results are observed for related data augmentation samplers for multinomial logit and probit models. When applied to a real quantitative advertising dataset, the data augmentation samplers mix very poorly. Conversely, Hamiltonian Monte Carlo and a type of independence chain Metropolis algorithm show good mixing on the same dataset.

Crystallographic and magnetic identification of secondary phase in orientated Bi 5Fe 0.5Co 0.5Ti 3O 15 ceramics

The fabrication of highly-oriented polycrystalline ceramics of Bi 5Fe 0.5Co 0.5Ti 3O 15, prepared via molten salt synthesis and uniaxial pressing of high aspect ratio platelets is reported. Electron backscatter images show a secondary phase within the ceramic which is rich in cobalt and iron. The concentration of the secondary phase obtained from scanning electron microscopy is estimated at less than 2% by volume, below the detection limit of x-ray diffraction (XRD). The samples were characterized by x-ray diffraction, polarization-electric field measurements, superconducting quantum interference device as a function of sample orientation and vibrating sample magnetometry as a function of temperature. It is inferred from the data that the observed ferromagnetic response is dominated by the secondary phase. This work highlights the importance of rigorous materials characterisation in the study of multiferroics as small amounts of secondary phase, below the limit of XRD, can lead to false conclusions.

Prayer and Depression: Women in Rural Pakistan

Background: Mental health, specifically depression, is a burden of disease in Pakistan. Religion and depression have not been studied in Pakistan currently, specially within a subset of a rural population. Methods: A secondary-data analysis was conducted using logistic regression for a non-parametrically distributed data set. The setting was in rural Pakistan, near Rawalpindi, and the sample size data was collected from the SHARE (South Asian Hub for Advocacy, Research, and Education). The measures used were the phq9 scaled for depression, prayer number, mother’s education, mother’s age, and if the mothers work. Results: This study demonstrated that there was no association between prayer and depression in this cohort. The mean prayer number between depressed and non-depressed women was 1.22 and 1.42, respectively, and a Wilcoxan rank sum test indicated that this was not significant. Conclusions: The primary finding indicates that increased frequency of prayer is not associated with a decreased rate of depression. This may be due to prayer number not being a significant enough measure. The implications of these findings stress the need for more depression intervention in rural Pakistan.

Symptom burden and its relationship to functional performance in patients with chronic obstructive pulmonary disease

The aim of this study was to explore symptom burden and its relationship to functional performance in patients with COPD. A descriptive, cross-sectional, correlational survey design was used and a sample of 214 patients with COPD. The sample was recruited from patients attending one of the major teaching hospitals in Dublin. Symptom burden was measured using the Memorial Symptom Assessment Scale (MSAS), and the functional performance was measured using the Functional Performance Inventory-Short Form (FPISF). Findings revealed that participants experienced a median of 13 symptoms. The most burdensome symptoms were shortness of breath, lack of energy, difficulty sleeping, worrying, dry mouth, feeling nervous, feeling irritable, and feeling sad. Participants with very severe COPD had the greatest symptom burden, followed by those with severe COPD, moderate COPD, and mild COPD. Symptom burden was higher for the psychological symptoms compared to the physical symptoms. Participants with mild COPD had the highest functional performance, followed by those with moderate COPD, very severe COPD, and severe COPD. Twenty symptoms were negatively correlated with overall functional performance, indicating that high symptom burden for those symptoms was associated with low overall functional performance. Moderate, negative, statistically significant correlations were found between the total symptom burden and overall functional performance, physical symptom burden and overall functional performance and psychological symptom burden and overall functional performance. A negative linear relationship was found between total symptom burden and overall functional performance among all stages of COPD except the mild group. No relationship was found between total symptom burden and overall functional performance for the moderate group. Healthcare professionals need to broaden the clinical and research assessment of physical and psychological symptoms in COPD; alleviating the burden of these symptoms may promote improved functional performance.

The Efficacy of San Lazaro and His Manifestations: Divine Mediators of Health Within Miami's Cuban-American Santeria Community.

This study explored the origins, evolution and influence of the tradition of San Lázaro as it currently pertains to the Cuban-American Santeria community in Miami. The main argument of the study is that in the context of the contemporary religious culture of Santeria in Miami, San Lázaro is a hybrid spirit. Many manifestations of healing entities have come to merge in the person of this spirit. Though practitioners identify with specific manifestations of this spirit, the processes of transmigration have blurred the lines of deep-rooted faiths and created a fusion of meanings from disparate traditions, making San Lázaro an ambivalent personality. San Lázaro’s ambivalence is the very quality that makes him such an important Orisha. As a deity whose personalities demonstrates the combination of a diversity of qualities, including those that contradict each other, San Lázaro is deployed in a very broad range of healing context, making him a versatile Orisha. This study clarified the contrasting qualities this deity embodies and traces the socio-historical context in which the deity acquires the layers of meanings it is currently associated with. Drawing on interviews with Lázaranian worshipers [Lázarenos] in Miami and engaging in Bourdieu’s concept of Habitus, the study provided a window into the nature of the tradition of San Lázaro and how its usage is linked with the African heritage of the worshipers.

Modulation of CYP1A1 by PKC Inhibitors and TPA Pre-Treatments in MH1C1 Rat Hepatoma Cells Exposed to 3 -Methylcholanthrene

Cytochrome P4501A1 (CYP1A1), an enzyme known to metabolize polycyclic aromatic hydrocarbons, is regulated by the aryl hydrocarbon receptor (AhR). The involvement of protein kinase C (PKC) in the regulation of AhR signal transduction pathway, has been widely studied but the role of specific PKC isoform(s) involved in this process it is not well clarified. To study which PKC isoform(s) is implicated in the regulation of CYP1A1, in the poorly tumorigenic MH1C1 rat hepatoma cells, we examined the effects of some PKC pharmacological inhibitors, Calphostin C (CAL), Staurosporine (STA) and H7, and of 12-0-tetradecanoyl phorbol 13-acetate (TPA), a PKC activator, on basal and 3- methylcholanthrene (MC)-induced CYP1A1 protein expression and mediated ethoxyresorufin O-deethylation (EROD) activity. In parallel, the activities of PKC-α, -βI, -δ and -ε isoforms, the most expressed in MH1C1 cells, were monitored. After pre-treatment with CAL, STA and H7, the MC-induced CYP1A1 protein and EROD activity were rapidly reduced with temporal profile similar to the profile of the activity of α and β1 PKC isoforms. Moreover, TPA pre-treatment induced a biphasic effect on EROD activity, and a decline of PKC -βI and -α, in first instance, and -δ and -ε activities later on. These findings clearly show that, in MH1C1 cells, PKC is involved in CYP1A1 regulation and that α and βI classic PKC isoforms play an active role in modulating this process.

Angiomatoid fibrous histiocytoma: comparison of fluorescence in situ hybridization and reverse transcription polymerase chain reaction as adjunct diagnostic modalities

Angiomatoid fibrous histiocytoma (AFH) is a rare soft tissue neoplasm of intermediate biologic potential and uncertain differentiation, most often arising in the extremities of children and young adults. Although it has characteristic histologic features of a lymphoid cuff surrounding nodules of ovoid cells with blood-filled cystic cavities, diagnosis is often difficult due to its morphologic heterogeneity and lack of specific immunoprofile. Angiomatoid fibrous histiocytoma is associated with recurrent chromosomal translocations, leading to characteristic EWSR1-CREB1, EWSR1-ATF1, and, rarely, FUS-ATF1 gene fusions; fluorescence in situ hybridization (FISH), detecting EWSR1 or FUS rearrangements, and reverse transcription-polymerase chain reaction (RT-PCR) for EWSR1-CREB1 and EWSR1-ATF1 fusion transcripts have become routine ancillary tools. We present a large comparative series of FISH and RT-PCR for AFH. Seventeen neoplasms (from 16 patients) histologically diagnosed as AFH were assessed for EWSR1 rearrangements or EWSR1-CREB1 and EWSR1-ATF1 fusion transcripts. All 17 were positive for either FISH or RT-PCR or both. Of 16, 14 (87.5%) had detectable EWSR1-CREB1 or EWSR1-ATF1 fusion transcripts by RT-PCR, whereas 13 (76.5%) of 17 had positive EWSR1 rearrangement with FISH. All 13 of 13 non-AFH control neoplasms failed to show EWSR1-CREB1 or EWSR1-ATF1 fusion transcripts, whereas EWSR1 rearrangement was present in 2 of these 13 cases (which were histopathologically myoepithelial neoplasms). This study shows that EWSR1-CREB1 or EWSR1-ATF1 fusions predominate in AFH (supporting previous reports that FUS rearrangement is rare in AFH) and that RT-PCR has a comparable detection rate to FISH for AFH. Importantly, cases of AFH can be missed if RT-PCR is not performed in conjunction with FISH, and RT-PCR has the added advantage of specificity, which is crucial, as EWSR1 rearrangements are present in a variety of neoplasms in the histologic differential diagnosis of AFH, that differ in behavior and treatment.