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.

Health events and the smoking cessation of middle aged Americans.

This study investigates the effect of serious health events including new diagnoses of heart attacks, strokes, cancers, chronic lung disease, chronic heart failure, diabetes, and heart disease on future smoking status up to 6 years postevent. Data come from the Health and Retirement Study, a nationally representative longitudinal survey of Americans aged 51-61 in 1991, followed every 2 years from 1992 to 1998. Smoking status is evaluated at each of three follow-ups, (1994, 1996, and 1998) as a function of health events between each of the four waves. Acute and chronic health events are associated with much lower likelihood of smoking both in the wave immediately following the event and up to 6 years later. However, future events do not retrospectively predict past cessation. In sum, serious health events have substantial impacts on cessation rates of older smokers. Notably, these effects persist for as much as 6 years after a health event.

Attenuation of inflammatory events in human intervertebral disc cells with a tumor necrosis factor antagonist.

STUDY DESIGN: The inflammatory responses of primary human intervertebral disc (IVD) cells to tumor necrosis factor α (TNF-α) and an antagonist were evaluated in vitro. OBJECTIVE: To investigate an ability for soluble TNF receptor type II (sTNFRII) to antagonize TNF-α-induced inflammatory events in primary human IVD cells in vitro. SUMMARY OF BACKGROUND DATA: TNF-α is a known mediator of inflammation and pain associated with radiculopathy and IVD degeneration. sTNFRs and their analogues are of interest for the clinical treatment of these IVD pathologies, although information on the effects of sTNFR on human IVD cells remains unknown. METHODS: IVD cells were isolated from surgical tissues procured from 15 patients and cultured with or without 1.4 nmol/L TNF-α (25 ng/mL). Treatment groups were coincubated with varying doses of sTNFRII (12.5-100 nmol/L). Nitric oxide (NO), prostaglandin E₂ (PGE₂), and interleukin-6 (IL6) levels in media were quantified to characterize the inflammatory phenotype of the IVD cells. RESULTS: Across all patients, TNF-α induced large, statistically significant increases in NO, PGE₂, and IL6 secretion from IVD cells compared with controls (60-, 112-, and 4-fold increases, respectively; P < 0.0001). Coincubation of TNF-α with nanomolar doses of sTNFRII significantly attenuated the secretion of NO and PGE₂ in a dose-dependent manner, whereas IL6 levels were unchanged. Mean IC₅₀ values for NO and PGE₂ were found to be 35.1 and 20.5 nmol/L, respectively. CONCLUSION: Nanomolar concentrations of sTNFRII were able to significantly attenuate the effects of TNF-α on primary human IVD cells in vitro. These results suggest this sTNFR to be a potent TNF antagonist with potential to attenuate inflammation in IVD pathology.

Rare variants create synthetic genome-wide associations.

Genome-wide association studies (GWAS) have now identified at least 2,000 common variants that appear associated with common diseases or related traits (http://www.genome.gov/gwastudies), hundreds of which have been convincingly replicated. It is generally thought that the associated markers reflect the effect of a nearby common (minor allele frequency >0.05) causal site, which is associated with the marker, leading to extensive resequencing efforts to find causal sites. We propose as an alternative explanation that variants much less common than the associated one may create "synthetic associations" by occurring, stochastically, more often in association with one of the alleles at the common site versus the other allele. Although synthetic associations are an obvious theoretical possibility, they have never been systematically explored as a possible explanation for GWAS findings. Here, we use simple computer simulations to show the conditions under which such synthetic associations will arise and how they may be recognized. We show that they are not only possible, but inevitable, and that under simple but reasonable genetic models, they are likely to account for or contribute to many of the recently identified signals reported in genome-wide association studies. We also illustrate the behavior of synthetic associations in real datasets by showing that rare causal mutations responsible for both hearing loss and sickle cell anemia create genome-wide significant synthetic associations, in the latter case extending over a 2.5-Mb interval encompassing scores of "blocks" of associated variants. In conclusion, uncommon or rare genetic variants can easily create synthetic associations that are credited to common variants, and this possibility requires careful consideration in the interpretation and follow up of GWAS signals.

Association of a peripheral blood metabolic profile with coronary artery disease and risk of subsequent cardiovascular events.

BACKGROUND: Molecular tools may provide insight into cardiovascular risk. We assessed whether metabolites discriminate coronary artery disease (CAD) and predict risk of cardiovascular events. METHODS AND RESULTS: We performed mass-spectrometry-based profiling of 69 metabolites in subjects from the CATHGEN biorepository. To evaluate discriminative capabilities of metabolites for CAD, 2 groups were profiled: 174 CAD cases and 174 sex/race-matched controls ("initial"), and 140 CAD cases and 140 controls ("replication"). To evaluate the capability of metabolites to predict cardiovascular events, cases were combined ("event" group); of these, 74 experienced death/myocardial infarction during follow-up. A third independent group was profiled ("event-replication" group; n=63 cases with cardiovascular events, 66 controls). Analysis included principal-components analysis, linear regression, and Cox proportional hazards. Two principal components analysis-derived factors were associated with CAD: 1 comprising branched-chain amino acid metabolites (factor 4, initial P=0.002, replication P=0.01), and 1 comprising urea cycle metabolites (factor 9, initial P=0.0004, replication P=0.01). In multivariable regression, these factors were independently associated with CAD in initial (factor 4, odds ratio [OR], 1.36; 95% CI, 1.06 to 1.74; P=0.02; factor 9, OR, 0.67; 95% CI, 0.52 to 0.87; P=0.003) and replication (factor 4, OR, 1.43; 95% CI, 1.07 to 1.91; P=0.02; factor 9, OR, 0.66; 95% CI, 0.48 to 0.91; P=0.01) groups. A factor composed of dicarboxylacylcarnitines predicted death/myocardial infarction (event group hazard ratio 2.17; 95% CI, 1.23 to 3.84; P=0.007) and was associated with cardiovascular events in the event-replication group (OR, 1.52; 95% CI, 1.08 to 2.14; P=0.01). CONCLUSIONS: Metabolite profiles are associated with CAD and subsequent cardiovascular events.

Souvenirs of conquest: Israeli occupations as tourist events

It is perhaps self-evident to suggest that military conquest shares something with tourism because both involve encounters with "strange" landscapes and people. Thus it may not surprise that the former sometimes borrows rhetorical strategies from the latter - strategies for rendering the strange familiar or for translating threatening images into benign ones. There have been numerous studies of this history of borrowing. Scholars have considered how scenes of battle draw tourist crowds, how soldiers' ways of seeing can resemble those of leisure travelers, how televised wars have been visually structured as tourist events (e.g., the 2003 U.S. invasion of Iraq), and how the spoils of war can function as a body of souvenirs. These lines of inquiry expand our understanding of tourism as a field of cultural practices and help us to rethink the parameters of militarism and warfare by suggesting ways they are entangled with everyday leisure practices. © 2008 Cambridge University Press.

Rare targets are rarely missed in correctable search.

Failing to find a tumor in an x-ray scan or a gun in an airport baggage screening can have dire consequences, making it fundamentally important to elucidate the mechanisms that hinder performance in such visual searches. Recent laboratory work has indicated that low target prevalence can lead to disturbingly high miss rates in visual search. Here, however, we demonstrate that misses in low-prevalence searches can be readily abated. When targets are rarely present, observers adapt by responding more quickly, and miss rates are high. Critically, though, these misses are often due to response-execution errors, not perceptual or identification errors: Observers know a target was present, but just respond too quickly. When provided an opportunity to correct their last response, observers can catch their mistakes. Thus, low target prevalence may not be a generalizable cause of high miss rates in visual search.

A functional polymorphism in the 5HTR2C gene associated with stress responses also predicts incident cardiovascular events.

Previously we have shown that a functional nonsynonymous single nucleotide polymorphism (rs6318) of the 5HTR2C gene located on the X-chromosome is associated with hypothalamic-pituitary-adrenal axis response to a stress recall task, and with endophenotypes associated with cardiovascular disease (CVD). These findings suggest that individuals carrying the rs6318 Ser23 C allele will be at higher risk for CVD compared to Cys23 G allele carriers. The present study examined allelic variation in rs6318 as a predictor of coronary artery disease (CAD) severity and a composite endpoint of all-cause mortality or myocardial infarction (MI) among Caucasian participants consecutively recruited through the cardiac catheterization laboratory at Duke University Hospital (Durham, NC) as part of the CATHGEN biorepository. Study population consisted of 6,126 Caucasian participants (4,036 [65.9%] males and 2,090 [34.1%] females). A total of 1,769 events occurred (1,544 deaths and 225 MIs; median follow-up time = 5.3 years, interquartile range = 3.3-8.2). Unadjusted Cox time-to-event regression models showed, compared to Cys23 G carriers, males hemizygous for Ser23 C and females homozygous for Ser23C were at increased risk for the composite endpoint of all-cause death or MI: Hazard Ratio (HR) = 1.47, 95% confidence interval (CI) = 1.17, 1.84, p = .0008. Adjusting for age, rs6318 genotype was not related to body mass index, diabetes, hypertension, dyslipidemia, smoking history, number of diseased coronary arteries, or left ventricular ejection fraction in either males or females. After adjustment for these covariates the estimate for the two Ser23 C groups was modestly attenuated, but remained statistically significant: HR = 1.38, 95% CI = 1.10, 1.73, p = .005. These findings suggest that this functional polymorphism of the 5HTR2C gene is associated with increased risk for CVD mortality and morbidity, but this association is apparently not explained by the association of rs6318 with traditional risk factors or conventional markers of atherosclerotic disease.

Schema-driven construction of future autobiographical traumatic events: the future is much more troubling than the past.

Research on future episodic thought has produced compelling theories and results in cognitive psychology, cognitive neuroscience, and clinical psychology. In experiments aimed to integrate these with basic concepts and methods from autobiographical memory research, 76 undergraduates remembered past and imagined future positive and negative events that had or would have a major impact on them. Correlations of the online ratings of visual and auditory imagery, emotion, and other measures demonstrated that individuals used the same processes to the same extent to remember past and construct future events. These measures predicted the theoretically important metacognitive judgment of past reliving and future "preliving" in similar ways. On standardized tests of reactions to traumatic events, scores for future negative events were much higher than scores for past negative events. The scores for future negative events were in the range that would qualify for a diagnosis of posttraumatic stress disorder (PTSD); the test was replicated (n = 52) to check for order effects. Consistent with earlier work, future events had less sensory vividness. Thus, the imagined symptoms of future events were unlikely to be caused by sensory vividness. In a second experiment, to confirm this, 63 undergraduates produced numerous added details between 2 constructions of the same negative future events; deficits in rated vividness were removed with no increase in the standardized tests of reactions to traumatic events. Neuroticism predicted individuals' reactions to negative past events but did not predict imagined reactions to future events. This set of novel methods and findings is interpreted in the contexts of the literatures of episodic future thought, autobiographical memory, PTSD, and classic schema theory.

Cumulative exposure to traumatic events in older adults.

OBJECTIVES: The present study examined the impact of cumulative trauma exposure on current posttraumatic stress disorder (PTSD) symptom severity in a nonclinical sample of adults in their 60s. The predictive utility of cumulative trauma exposure was compared to other known predictors of PTSD, including trauma severity, personality traits, social support, and event centrality. METHOD: Community-dwelling adults (n = 2515) from the crest of the Baby Boom generation completed the Traumatic Life Events Questionnaire, the PTSD Checklist, the NEO Personality Inventory, the Centrality of Event Scale, and rated their current social support. RESULTS: Cumulative trauma exposure predicted greater PTSD symptom severity in hierarchical regression analyses consistent with a dose-response model. Neuroticism and event centrality also emerged as robust predictors of PTSD symptom severity. In contrast, the severity of individuals' single most distressing life event, as measured by self-report ratings of the A1 PTSD diagnostic criterion, did not add explanatory variance to the model. Analyses concerning event categories revealed that cumulative exposure to childhood violence and adulthood physical assaults were most strongly associated with PTSD symptom severity in older adulthood. Moreover, cumulative self-oriented events accounted for a larger percentage of variance in symptom severity compared to events directed at others. CONCLUSION: Our findings suggest that the cumulative impact of exposure to traumatic events throughout the life course contributes significantly to posttraumatic stress in older adulthood above and beyond other known predictors of PTSD.

Measuring the Severity of Negative and Traumatic Events.

We devised three measures of the general severity of events, which raters applied to participants' narrative descriptions: 1) placing events on a standard normed scale of stressful events, 2) placing events into five bins based on their severity relative to all other events in the sample, and 3) an average of ratings of the events' effects on six distinct areas of the participants' lives. Protocols of negative events were obtained from two non-diagnosed undergraduate samples (n = 688 and 328), a clinically diagnosed undergraduate sample all of whom had traumas and half of whom met PTSD criteria (n = 30), and a clinically diagnosed community sample who met PTSD criteria (n = 75). The three measures of severity correlated highly in all four samples but failed to correlate with PTSD symptom severity in any sample. Theoretical implications for the role of trauma severity in PTSD are discussed.

The Frequency and Impact of Exposure to Potentially Traumatic Events Over the Life Course.

We examined the frequency and impact of exposure to potentially traumatic events among a nonclinical sample of older adults (n = 3,575), a population typically underrepresented in epidemiological research concerning the prevalence of traumatic events. Current PTSD symptom severity and the centrality of events to identity were assessed for events nominated as currently most distressing. Approximately 90% of participants experienced one or more potentially traumatic events. Events that occurred with greater frequency early in the life course were associated with more severe PTSD symptoms compared to events that occurred with greater frequency during later decades. Early life traumas, however, were not more central to identity. Results underscore the differential impact of traumatic events experienced throughout the life course. We conclude with suggestions for further research concerning mechanisms that promote the persistence of post-traumatic stress related to early life traumas and empirical evaluation of psychotherapeutic treatments for older adults with PTSD.

Autobiographical memory for stressful events: the role of autobiographical memory in posttraumatic stress disorder.

To provide the three-way comparisons needed to test existing theories, we compared (1) most-stressful memories to other memories and (2) involuntary to voluntary memories (3) in 75 community dwelling adults with and 42 without a current diagnosis of posttraumatic stress disorder (PTSD). Each rated their three most-stressful, three most-positive, seven most-important and 15 word-cued autobiographical memories, and completed tests of personality and mood. Involuntary memories were then recorded and rated as they occurred for 2 weeks. Standard mechanisms of cognition and affect applied to extreme events accounted for the properties of stressful memories. Involuntary memories had greater emotional intensity than voluntary memories, but were not more frequently related to traumatic events. The emotional intensity, rehearsal, and centrality to the life story of both voluntary and involuntary memories, rather than incoherence of voluntary traumatic memories and enhanced availability of involuntary traumatic memories, were the properties of autobiographical memories associated with PTSD.