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.

Phylotastic! Making tree-of-life knowledge accessible, reusable and convenient.

BACKGROUND: Scientists rarely reuse expert knowledge of phylogeny, in spite of years of effort to assemble a great "Tree of Life" (ToL). A notable exception involves the use of Phylomatic, which provides tools to generate custom phylogenies from a large, pre-computed, expert phylogeny of plant taxa. This suggests great potential for a more generalized system that, starting with a query consisting of a list of any known species, would rectify non-standard names, identify expert phylogenies containing the implicated taxa, prune away unneeded parts, and supply branch lengths and annotations, resulting in a custom phylogeny suited to the user's needs. Such a system could become a sustainable community resource if implemented as a distributed system of loosely coupled parts that interact through clearly defined interfaces. RESULTS: With the aim of building such a "phylotastic" system, the NESCent Hackathons, Interoperability, Phylogenies (HIP) working group recruited 2 dozen scientist-programmers to a weeklong programming hackathon in June 2012. During the hackathon (and a three-month follow-up period), 5 teams produced designs, implementations, documentation, presentations, and tests including: (1) a generalized scheme for integrating components; (2) proof-of-concept pruners and controllers; (3) a meta-API for taxonomic name resolution services; (4) a system for storing, finding, and retrieving phylogenies using semantic web technologies for data exchange, storage, and querying; (5) an innovative new service, DateLife.org, which synthesizes pre-computed, time-calibrated phylogenies to assign ages to nodes; and (6) demonstration projects. These outcomes are accessible via a public code repository (GitHub.com), a website (http://www.phylotastic.org), and a server image. CONCLUSIONS: Approximately 9 person-months of effort (centered on a software development hackathon) resulted in the design and implementation of proof-of-concept software for 4 core phylotastic components, 3 controllers, and 3 end-user demonstration tools. While these products have substantial limitations, they suggest considerable potential for a distributed system that makes phylogenetic knowledge readily accessible in computable form. Widespread use of phylotastic systems will create an electronic marketplace for sharing phylogenetic knowledge that will spur innovation in other areas of the ToL enterprise, such as annotation of sources and methods and third-party methods of quality assessment.

Identifying the Structure and Fate of Wastewater Derived Organic Micropollutants by High-resolution Mass Spectrometry

Human activities represent a significant burden on the global water cycle, with large and increasing demands placed on limited water resources by manufacturing, energy production and domestic water use. In addition to changing the quantity of available water resources, human activities lead to changes in water quality by introducing a large and often poorly-characterized array of chemical pollutants, which may negatively impact biodiversity in aquatic ecosystems, leading to impairment of valuable ecosystem functions and services. Domestic and industrial wastewaters represent a significant source of pollution to the aquatic environment due to inadequate or incomplete removal of chemicals introduced into waters by human activities. Currently, incomplete chemical characterization of treated wastewaters limits comprehensive risk assessment of this ubiquitous impact to water. In particular, a significant fraction of the organic chemical composition of treated industrial and domestic wastewaters remains uncharacterized at the molecular level. Efforts aimed at reducing the impacts of water pollution on aquatic ecosystems critically require knowledge of the composition of wastewaters to develop interventions capable of protecting our precious natural water resources.

The goal of this dissertation was to develop a robust, extensible and high-throughput framework for the comprehensive characterization of organic micropollutants in wastewaters by high-resolution accurate-mass mass spectrometry. High-resolution mass spectrometry provides the most powerful analytical technique available for assessing the occurrence and fate of organic pollutants in the water cycle. However, significant limitations in data processing, analysis and interpretation have limited this technique in achieving comprehensive characterization of organic pollutants occurring in natural and built environments. My work aimed to address these challenges by development of automated workflows for the structural characterization of organic pollutants in wastewater and wastewater impacted environments by high-resolution mass spectrometry, and to apply these methods in combination with novel data handling routines to conduct detailed fate studies of wastewater-derived organic micropollutants in the aquatic environment.

In Chapter 2, chemoinformatic tools were implemented along with novel non-targeted mass spectrometric analytical methods to characterize, map, and explore an environmentally-relevant “chemical space” in municipal wastewater. This was accomplished by characterizing the molecular composition of known wastewater-derived organic pollutants and substances that are prioritized as potential wastewater contaminants, using these databases to evaluate the pollutant-likeness of structures postulated for unknown organic compounds that I detected in wastewater extracts using high-resolution mass spectrometry approaches. Results showed that application of multiple computational mass spectrometric tools to structural elucidation of unknown organic pollutants arising in wastewaters improved the efficiency and veracity of screening approaches based on high-resolution mass spectrometry. Furthermore, structural similarity searching was essential for prioritizing substances sharing structural features with known organic pollutants or industrial and consumer chemicals that could enter the environment through use or disposal.

I then applied this comprehensive methodological and computational non-targeted analysis workflow to micropollutant fate analysis in domestic wastewaters (Chapter 3), surface waters impacted by water reuse activities (Chapter 4) and effluents of wastewater treatment facilities receiving wastewater from oil and gas extraction activities (Chapter 5). In Chapter 3, I showed that application of chemometric tools aided in the prioritization of non-targeted compounds arising at various stages of conventional wastewater treatment by partitioning high dimensional data into rational chemical categories based on knowledge of organic chemical fate processes, resulting in the classification of organic micropollutants based on their occurrence and/or removal during treatment. Similarly, in Chapter 4, high-resolution sampling and broad-spectrum targeted and non-targeted chemical analysis were applied to assess the occurrence and fate of organic micropollutants in a water reuse application, wherein reclaimed wastewater was applied for irrigation of turf grass. Results showed that organic micropollutant composition of surface waters receiving runoff from wastewater irrigated areas appeared to be minimally impacted by wastewater-derived organic micropollutants. Finally, Chapter 5 presents results of the comprehensive organic chemical composition of oil and gas wastewaters treated for surface water discharge. Concurrent analysis of effluent samples by complementary, broad-spectrum analytical techniques, revealed that low-levels of hydrophobic organic contaminants, but elevated concentrations of polymeric surfactants, which may effect the fate and analysis of contaminants of concern in oil and gas wastewaters.

Taken together, my work represents significant progress in the characterization of polar organic chemical pollutants associated with wastewater-impacted environments by high-resolution mass spectrometry. Application of these comprehensive methods to examine micropollutant fate processes in wastewater treatment systems, water reuse environments, and water applications in oil/gas exploration yielded new insights into the factors that influence transport, transformation, and persistence of organic micropollutants in these systems across an unprecedented breadth of chemical space.

Adaptive Mixture Modelling Metropolis Methods for Bayesian Analysis of Non-linear State-Space Models.

We describe a strategy for Markov chain Monte Carlo analysis of non-linear, non-Gaussian state-space models involving batch analysis for inference on dynamic, latent state variables and fixed model parameters. The key innovation is a Metropolis-Hastings method for the time series of state variables based on sequential approximation of filtering and smoothing densities using normal mixtures. These mixtures are propagated through the non-linearities using an accurate, local mixture approximation method, and we use a regenerating procedure to deal with potential degeneracy of mixture components. This provides accurate, direct approximations to sequential filtering and retrospective smoothing distributions, and hence a useful construction of global Metropolis proposal distributions for simulation of posteriors for the set of states. This analysis is embedded within a Gibbs sampler to include uncertain fixed parameters. We give an example motivated by an application in systems biology. Supplemental materials provide an example based on a stochastic volatility model as well as MATLAB code.

Functional annotation signatures of disease susceptibility loci improve SNP association analysis.

BACKGROUND: Genetic association studies are conducted to discover genetic loci that contribute to an inherited trait, identify the variants behind these associations and ascertain their functional role in determining the phenotype. To date, functional annotations of the genetic variants have rarely played more than an indirect role in assessing evidence for association. Here, we demonstrate how these data can be systematically integrated into an association study's analysis plan. RESULTS: We developed a Bayesian statistical model for the prior probability of phenotype-genotype association that incorporates data from past association studies and publicly available functional annotation data regarding the susceptibility variants under study. The model takes the form of a binary regression of association status on a set of annotation variables whose coefficients were estimated through an analysis of associated SNPs in the GWAS Catalog (GC). The functional predictors examined included measures that have been demonstrated to correlate with the association status of SNPs in the GC and some whose utility in this regard is speculative: summaries of the UCSC Human Genome Browser ENCODE super-track data, dbSNP function class, sequence conservation summaries, proximity to genomic variants in the Database of Genomic Variants and known regulatory elements in the Open Regulatory Annotation database, PolyPhen-2 probabilities and RegulomeDB categories. Because we expected that only a fraction of the annotations would contribute to predicting association, we employed a penalized likelihood method to reduce the impact of non-informative predictors and evaluated the model's ability to predict GC SNPs not used to construct the model. We show that the functional data alone are predictive of a SNP's presence in the GC. Further, using data from a genome-wide study of ovarian cancer, we demonstrate that their use as prior data when testing for association is practical at the genome-wide scale and improves power to detect associations. CONCLUSIONS: We show how diverse functional annotations can be efficiently combined to create 'functional signatures' that predict the a priori odds of a variant's association to a trait and how these signatures can be integrated into a standard genome-wide-scale association analysis, resulting in improved power to detect truly associated variants.