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.

Exploring Mechanisms of Bacterial Adaptation to Seasonal Temperature Change

This research examines three potential mechanisms by which bacteria can adapt to different temperatures: changes in strain-level population structure, gene regulation and particle colonization. For the first two mechanisms, I utilize bacterial strains from the Vibrionaceae family due to their ease of culturability, ubiquity in coastal environments and status as a model system for marine bacteria. I first examine vibrio seasonal dynamics in temperate, coastal water and compare the thermal performance of strains that occupy different thermal environments. Our results suggest that there are tradeoffs in adaptation to specific temperatures and that thermal specialization can occur at a very fine phylogenetic scale. The observed thermal specialization over relatively short evolutionary time-scales indicates that few genes or cellular processes may limit expansion to a different thermal niche. I then compare the genomic and transcriptional changes associated with thermal adaptation in closely-related vibrio strains under heat and cold stress. The two vibrio strains have very similar genomes and overall exhibit similar transcriptional profiles in response to temperature stress but their temperature preferences are determined by differential transcriptional responses in shared genes as well as temperature-dependent regulation of unique genes. Finally, I investigate the temporal dynamics of particle-attached and free-living bacterial community in coastal seawater and find that microhabitats exert a stronger forcing on microbial communities than environmental variability, suggesting that particle-attachment could buffer the impacts of environmental changes and particle-associated communities likely respond to the presence of distinct eukaryotes rather than commonly-measured environmental parameters. Integrating these results will offer new perspectives on the mechanisms by which bacteria respond to seasonal temperature changes as well as potential adaptations to climate change-driven warming of the surface oceans.

Spatial Relationships among Hydroacoustic, Hydrographic and Top Predator Patterns: Cetacean Distributions in the Mid-Atlantic Bight

Effective conservation and management of top predators requires a comprehensive understanding of their distributions and of the underlying biological and physical processes that affect these distributions. The Mid-Atlantic Bight shelf break system is a dynamic and productive region where at least 32 species of cetaceans have been recorded through various systematic and opportunistic marine mammal surveys from the 1970s through 2012. My dissertation characterizes the spatial distribution and habitat of cetaceans in the Mid-Atlantic Bight shelf break system by utilizing marine mammal line-transect survey data, synoptic multi-frequency active acoustic data, and fine-scale hydrographic data collected during the 2011 summer Atlantic Marine Assessment Program for Protected Species (AMAPPS) survey. Although studies describing cetacean habitat and distributions have been previously conducted in the Mid-Atlantic Bight, my research specifically focuses on the shelf break region to elucidate both the physical and biological processes that influence cetacean distribution patterns within this cetacean hotspot.

In Chapter One I review biologically important areas for cetaceans in the Atlantic waters of the United States. I describe the study area, the shelf break region of the Mid-Atlantic Bight, in terms of the general oceanography, productivity and biodiversity. According to recent habitat-based cetacean density models, the shelf break region is an area of high cetacean abundance and density, yet little research is directed at understanding the mechanisms that establish this region as a cetacean hotspot.

In Chapter Two I present the basic physical principles of sound in water and describe the methodology used to categorize opportunistically collected multi-frequency active acoustic data using frequency responses techniques. Frequency response classification methods are usually employed in conjunction with net-tow data, but the logistics of the 2011 AMAPPS survey did not allow for appropriate net-tow data to be collected. Biologically meaningful information can be extracted from acoustic scattering regions by comparing the frequency response curves of acoustic regions to theoretical curves of known scattering models. Using the five frequencies on the EK60 system (18, 38, 70, 120, and 200 kHz), three categories of scatterers were defined: fish-like (with swim bladder), nekton-like (e.g., euphausiids), and plankton-like (e.g., copepods). I also employed a multi-frequency acoustic categorization method using three frequencies (18, 38, and 120 kHz) that has been used in the Gulf of Maine and Georges Bank which is based the presence or absence of volume backscatter above a threshold. This method is more objective than the comparison of frequency response curves because it uses an established backscatter value for the threshold. By removing all data below the threshold, only strong scattering information is retained.

In Chapter Three I analyze the distribution of the categorized acoustic regions of interest during the daytime cross shelf transects. Over all transects, plankton-like acoustic regions of interest were detected most frequently, followed by fish-like acoustic regions and then nekton-like acoustic regions. Plankton-like detections were the only significantly different acoustic detections per kilometer, although nekton-like detections were only slightly not significant. Using the threshold categorization method by Jech and Michaels (2006) provides a more conservative and discrete detection of acoustic scatterers and allows me to retrieve backscatter values along transects in areas that have been categorized. This provides continuous data values that can be integrated at discrete spatial increments for wavelet analysis. Wavelet analysis indicates significant spatial scales of interest for fish-like and nekton-like acoustic backscatter range from one to four kilometers and vary among transects.

In Chapter Four I analyze the fine scale distribution of cetaceans in the shelf break system of the Mid-Atlantic Bight using corrected sightings per trackline region, classification trees, multidimensional scaling, and random forest analysis. I describe habitat for common dolphins, Risso’s dolphins and sperm whales. From the distribution of cetacean sightings, patterns of habitat start to emerge: within the shelf break region of the Mid-Atlantic Bight, common dolphins were sighted more prevalently over the shelf while sperm whales were more frequently found in the deep waters offshore and Risso’s dolphins were most prevalent at the shelf break. Multidimensional scaling presents clear environmental separation among common dolphins and Risso’s dolphins and sperm whales. The sperm whale random forest habitat model had the lowest misclassification error (0.30) and the Risso’s dolphin random forest habitat model had the greatest misclassification error (0.37). Shallow water depth (less than 148 meters) was the primary variable selected in the classification model for common dolphin habitat. Distance to surface density fronts and surface temperature fronts were the primary variables selected in the classification models to describe Risso’s dolphin habitat and sperm whale habitat respectively. When mapped back into geographic space, these three cetacean species occupy different fine-scale habitats within the dynamic Mid-Atlantic Bight shelf break system.

In Chapter Five I present a summary of the previous chapters and present potential analytical steps to address ecological questions pertaining the dynamic shelf break region. Taken together, the results of my dissertation demonstrate the use of opportunistically collected data in ecosystem studies; emphasize the need to incorporate middle trophic level data and oceanographic features into cetacean habitat models; and emphasize the importance of developing more mechanistic understanding of dynamic ecosystems.