Predicting Transcript Production Rates in Yeast With Sparse Linear Models

To provide biological insights into transcriptional regulation, a couple of groups have recently presented models relating the promoter DNA-bound transcription factors (TFs) to downstream gene’s mean transcript level or transcript production rates over time. However, transcript production is dynamic in response to changes of TF concentrations over time. Also, TFs are not the only factors binding to promoters; other DNA binding factors (DBFs) bind as well, especially nucleosomes, resulting in competition between DBFs for binding at same genomic location. Additionally, not only TFs, but also some other elements regulate transcription. Within core promoter, various regulatory elements influence RNAPII recruitment, PIC formation, RNAPII searching for TSS, and RNAPII initiating transcription. Moreover, it is proposed that downstream from TSS, nucleosomes resist RNAPII elongation.

Here, we provide a machine learning framework to predict transcript production rates from DNA sequences. We applied this framework in the S. cerevisiae yeast for two scenarios: a) to predict the dynamic transcript production rate during the cell cycle for native promoters; b) to predict the mean transcript production rate over time for synthetic promoters. As far as we know, our framework is the first successful attempt to have a model that can predict dynamic transcript production rates from DNA sequences only: with cell cycle data set, we got Pearson correlation coefficient Cp = 0.751 and coefficient of determination r2 = 0.564 on test set for predicting dynamic transcript production rate over time. Also, for DREAM6 Gene Promoter Expression Prediction challenge, our fitted model outperformed all participant teams, best of all teams, and a model combining best team’s k-mer based sequence features and another paper’s biologically mechanistic features, in terms of all scoring metrics.

Moreover, our framework shows its capability of identifying generalizable fea- tures by interpreting the highly predictive models, and thereby provide support for associated hypothesized mechanisms about transcriptional regulation. With the learned sparse linear models, we got results supporting the following biological insights: a) TFs govern the probability of RNAPII recruitment and initiation possibly through interactions with PIC components and transcription cofactors; b) the core promoter amplifies the transcript production probably by influencing PIC formation, RNAPII recruitment, DNA melting, RNAPII searching for and selecting TSS, releasing RNAPII from general transcription factors, and thereby initiation; c) there is strong transcriptional synergy between TFs and core promoter elements; d) the regulatory elements within core promoter region are more than TATA box and nucleosome free region, suggesting the existence of still unidentified TAF-dependent and cofactor-dependent core promoter elements in yeast S. cerevisiae; e) nucleosome occupancy is helpful for representing +1 and -1 nucleosomes’ regulatory roles on transcription.

Mixtures of g-priors in Generalized Linear Models

Mixtures of Zellner's g-priors have been studied extensively in linear models and have been shown to have numerous desirable properties for Bayesian variable selection and model averaging. Several extensions of g-priors to Generalized Linear Models (GLMs) have been proposed in the literature; however, the choice of prior distribution of g and resulting properties for inference have received considerably less attention. In this paper, we extend mixtures of g-priors to GLMs by assigning the truncated Compound Confluent Hypergeometric (tCCH) distribution to 1/(1+g) and illustrate how this prior distribution encompasses several special cases of mixtures of g-priors in the literature, such as the Hyper-g, truncated Gamma, Beta-prime, and the Robust prior. Under an integrated Laplace approximation to the likelihood, the posterior distribution of 1/(1+g) is in turn a tCCH distribution, and approximate marginal likelihoods are thus available analytically. We discuss the local geometric properties of the g-prior in GLMs and show that specific choices of the hyper-parameters satisfy the various desiderata for model selection proposed by Bayarri et al, such as asymptotic model selection consistency, information consistency, intrinsic consistency, and measurement invariance. We also illustrate inference using these priors and contrast them to others in the literature via simulation and real examples.

Volume, volatility, and leverage: A dynamic analysis

This paper uses dynamic impulse response analysis to investigate the interrelationships among stock price volatility, trading volume, and the leverage effect. Dynamic impulse response analysis is a technique for analyzing the multi-step-ahead characteristics of a nonparametric estimate of the one-step conditional density of a strictly stationary process. The technique is the generalization to a nonlinear process of Sims-style impulse response analysis for linear models. In this paper, we refine the technique and apply it to a long panel of daily observations on the price and trading volume of four stocks actively traded on the NYSE: Boeing, Coca-Cola, IBM, and MMM.

Daily intake of antioxidants in relation to survival among adult patients diagnosed with malignant glioma.

BACKGROUND: Malignant glioma is a rare cancer with poor survival. The influence of diet and antioxidant intake on glioma survival is not well understood. The current study examines the association between antioxidant intake and survival after glioma diagnosis. METHODS: Adult patients diagnosed with malignant glioma during 1991-1994 and 1997-2001 were enrolled in a population-based study. Diagnosis was confirmed by review of pathology specimens. A modified food-frequency questionnaire interview was completed by each glioma patient or a designated proxy. Intake of each food item was converted to grams consumed/day. From this nutrient database, 16 antioxidants, calcium, a total antioxidant index and 3 macronutrients were available for survival analysis. Cox regression estimated mortality hazard ratios associated with each nutrient and the antioxidant index adjusting for potential confounders. Nutrient values were categorized into tertiles. Models were stratified by histology (Grades II, III, and IV) and conducted for all (including proxy) subjects and for a subset of self-reported subjects. RESULTS: Geometric mean values for 11 fat-soluble and 6 water-soluble individual antioxidants, antioxidant index and 3 macronutrients were virtually the same when comparing all cases (n=748) to self-reported cases only (n=450). For patients diagnosed with Grade II and Grade III histology, moderate (915.8-2118.3 mcg) intake of fat-soluble lycopene was associated with poorer survival when compared to low intake (0.0-914.8 mcg), for self-reported cases only. High intake of vitamin E and moderate/high intake of secoisolariciresinol among Grade III patients indicated greater survival for all cases. In Grade IV patients, moderate/high intake of cryptoxanthin and high intake of secoisolariciresinol were associated with poorer survival among all cases. Among Grade II patients, moderate intake of water-soluble folate was associated with greater survival for all cases; high intake of vitamin C and genistein and the highest level of the antioxidant index were associated with poorer survival for all cases. CONCLUSIONS: The associations observed in our study suggest that the influence of some antioxidants on survival following a diagnosis of malignant glioma are inconsistent and vary by histology group. Further research in a large sample of glioma patients is needed to confirm/refute our results.

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.

Chimpanzees and bonobos exhibit divergent spatial memory development.

Spatial cognition and memory are critical cognitive skills underlying foraging behaviors for all primates. While the emergence of these skills has been the focus of much research on human children, little is known about ontogenetic patterns shaping spatial cognition in other species. Comparative developmental studies of nonhuman apes can illuminate which aspects of human spatial development are shared with other primates, versus which aspects are unique to our lineage. Here we present three studies examining spatial memory development in our closest living relatives, chimpanzees (Pan troglodytes) and bonobos (P. paniscus). We first compared memory in a naturalistic foraging task where apes had to recall the location of resources hidden in a large outdoor enclosure with a variety of landmarks (Studies 1 and 2). We then compared older apes using a matched memory choice paradigm (Study 3). We found that chimpanzees exhibited more accurate spatial memory than bonobos across contexts, supporting predictions from these species' different feeding ecologies. Furthermore, chimpanzees - but not bonobos - showed developmental improvements in spatial memory, indicating that bonobos exhibit cognitive paedomorphism (delays in developmental timing) in their spatial abilities relative to chimpanzees. Together, these results indicate that the development of spatial memory may differ even between closely related species. Moreover, changes in the spatial domain can emerge during nonhuman ape ontogeny, much like some changes seen in human children.

The missense of smell: functional variability in the human odorant receptor repertoire.

Humans have ~400 intact odorant receptors, but each individual has a unique set of genetic variations that lead to variation in olfactory perception. We used a heterologous assay to determine how often genetic polymorphisms in odorant receptors alter receptor function. We identified agonists for 18 odorant receptors and found that 63% of the odorant receptors we examined had polymorphisms that altered in vitro function. On average, two individuals have functional differences at over 30% of their odorant receptor alleles. To show that these in vitro results are relevant to olfactory perception, we verified that variations in OR10G4 genotype explain over 15% of the observed variation in perceived intensity and over 10% of the observed variation in perceived valence for the high-affinity in vitro agonist guaiacol but do not explain phenotype variation for the lower-affinity agonists vanillin and ethyl vanillin.

Victims of infanticide and conspecific bite wounding in a female-dominant primate: a long-term study.

The aggression animals receive from conspecifics varies between individuals across their lifetime. As poignantly evidenced by infanticide, for example, aggression can have dramatic fitness consequences. Nevertheless, we understand little about the sources of variation in received aggression, particularly in females. Using a female-dominant species renowned for aggressivity in both sexes, we tested for potential social, demographic, and genetic patterns in the frequency with which animals were wounded by conspecifics. Our study included 243 captive, ring-tailed lemurs (Lemur catta), followed from infancy to adulthood over a 35-year time span. We extracted injury, social, and life-history information from colony records and calculated neutral heterozygosity for a subset of animals, as an estimate of genetic diversity. Focusing on victims rather than aggressors, we used General Linear Models to explain bite-wound patterns at different life stages. In infancy, maternal age best predicted wounds received, as infants born to young mothers were the most frequent infanticide victims. In adulthood, sex best predicted wounds received, as males were three times more likely than females to be seriously injured. No relation emerged between wounds received and the other variables studied. Beyond the generally expected costs of adult male intrasexual aggression, we suggest possible additive costs associated with female-dominant societies - those suffered by young mothers engaged in aggressive disputes and those suffered by adult males aggressively targeted by both sexes. We propose that infanticide in lemurs may be a costly by-product of aggressively mediated, female social dominance. Accordingly, the benefits of female behavioral 'masculinization' accrued to females through priority of access to resources, may be partially offset by early costs in reproductive success. Understanding the factors that influence lifetime patterns of conspecific wounding is critical to evaluating the fitness costs associated with social living; however, these costs may vary substantially between societies.

Reduced length of hospital stay in colorectal surgery after implementation of an enhanced recovery protocol.

BACKGROUND: Enhanced recovery after surgery (ERAS) is a multimodal approach to perioperative care that combines a range of interventions to enable early mobilization and feeding after surgery. We investigated the feasibility, clinical effectiveness, and cost savings of an ERAS program at a major U. S. teaching hospital. METHODS: Data were collected from consecutive patients undergoing open or laparoscopic colorectal surgery during 2 time periods, before and after implementation of an ERAS protocol. Data collected included patient demographics, operative, and perioperative surgical and anesthesia data, need for analgesics, complications, inpatient medical costs, and 30-day readmission rates. RESULTS: There were 99 patients in the traditional care group, and 142 in the ERAS group. The median length of stay (LOS) was 5 days in the ERAS group compared with 7 days in the traditional group (P < 0.001). The reduction in LOS was significant for both open procedures (median 6 vs 7 days, P = 0.01), and laparoscopic procedures (4 vs 6 days, P < 0.0001). ERAS patients had fewer urinary tract infections (13% vs 24%, P = 0.03). Readmission rates were lower in ERAS patients (9.8% vs 20.2%, P = 0.02). DISCUSSION: Implementation of an enhanced recovery protocol for colorectal surgery at a tertiary medical center was associated with a significantly reduced LOS and incidence of urinary tract infection. This is consistent with that of other studies in the literature and suggests that enhanced recovery programs could be implemented successfully and should be considered in U.S. hospitals.

Energy recovery in individuals with knee osteoarthritis.

OBJECTIVE: Pathological gaits have been shown to limit transfer between potential (PE) and kinetic (KE) energy during walking, which can increase locomotor costs. The purpose of this study was to examine whether energy exchange would be limited in people with knee osteoarthritis (OA). METHODS: Ground reaction forces during walking were collected from 93 subjects with symptomatic knee OA (self-selected and fast speeds) and 13 healthy controls (self-selected speed) and used to calculate their center of mass (COM) movements, PE and KE relationships, and energy recovery during a stride. Correlations and linear regressions examined the impact of energy fluctuation phase and amplitude, walking velocity, body mass, self-reported pain, and radiographic severity on recovery. Paired t-tests were run to compare energy recovery between cohorts. RESULTS: Symptomatic knee OA subjects displayed lower energetic recovery during self-selected walking speeds than healthy controls (P = 0.0018). PE and KE phase relationships explained the majority (66%) of variance in recovery. Recovery had a complex relationship with velocity and its change across speeds was significantly influenced by the self-selected walking speed of each subject. Neither radiographic OA scores nor subject self-reported measures demonstrated any relationship with energy recovery. CONCLUSIONS: Knee OA reduces effective exchange of PE and KE, potentially increasing the muscular work required to control movements of the COM. Gait retraining may return subjects to more normal patterns of energy exchange and allow them to reduce fatigue.

Long-term dynamics of death rates of emphysema, asthma, and pneumonia and improving air quality.

BACKGROUND: The respiratory tract is a major target of exposure to air pollutants, and respiratory diseases are associated with both short- and long-term exposures. We hypothesized that improved air quality in North Carolina was associated with reduced rates of death from respiratory diseases in local populations. MATERIALS AND METHODS: We analyzed the trends of emphysema, asthma, and pneumonia mortality and changes of the levels of ozone, sulfur dioxide (SO2), nitrogen dioxide (NO2), carbon monoxide (CO), and particulate matters (PM2.5 and PM10) using monthly data measurements from air-monitoring stations in North Carolina in 1993-2010. The log-linear model was used to evaluate associations between air-pollutant levels and age-adjusted death rates (per 100,000 of population) calculated for 5-year age-groups and for standard 2000 North Carolina population. The studied associations were adjusted by age group-specific smoking prevalence and seasonal fluctuations of disease-specific respiratory deaths. RESULTS: Decline in emphysema deaths was associated with decreasing levels of SO2 and CO in the air, decline in asthma deaths-with lower SO2, CO, and PM10 levels, and decline in pneumonia deaths-with lower levels of SO2. Sensitivity analyses were performed to study potential effects of the change from International Classification of Diseases (ICD)-9 to ICD-10 codes, the effects of air pollutants on mortality during summer and winter, the impact of approach when only the underlying causes of deaths were used, and when mortality and air-quality data were analyzed on the county level. In each case, the results of sensitivity analyses demonstrated stability. The importance of analysis of pneumonia as an underlying cause of death was also highlighted. CONCLUSION: Significant associations were observed between decreasing death rates of emphysema, asthma, and pneumonia and decreases in levels of ambient air pollutants in North Carolina.

Credit scores, cardiovascular disease risk, and human capital.

Credit scores are the most widely used instruments to assess whether or not a person is a financial risk. Credit scoring has been so successful that it has expanded beyond lending and into our everyday lives, even to inform how insurers evaluate our health. The pervasive application of credit scoring has outpaced knowledge about why credit scores are such useful indicators of individual behavior. Here we test if the same factors that lead to poor credit scores also lead to poor health. Following the Dunedin (New Zealand) Longitudinal Study cohort of 1,037 study members, we examined the association between credit scores and cardiovascular disease risk and the underlying factors that account for this association. We find that credit scores are negatively correlated with cardiovascular disease risk. Variation in household income was not sufficient to account for this association. Rather, individual differences in human capital factors—educational attainment, cognitive ability, and self-control—predicted both credit scores and cardiovascular disease risk and accounted for ∼45% of the correlation between credit scores and cardiovascular disease risk. Tracing human capital factors back to their childhood antecedents revealed that the characteristic attitudes, behaviors, and competencies children develop in their first decade of life account for a significant portion (∼22%) of the link between credit scores and cardiovascular disease risk at midlife. We discuss the implications of these findings for policy debates about data privacy, financial literacy, and early childhood interventions.

Variation in pediatric traumatic brain injury outcomes in the United States.

OBJECTIVE: To ascertain the degree of variation, by state of hospitalization, in outcomes associated with traumatic brain injury (TBI) in a pediatric population. DESIGN: A retrospective cohort study of pediatric patients admitted to a hospital with a TBI. SETTING: Hospitals from states in the United States that voluntarily participate in the Agency for Healthcare Research and Quality's Healthcare Cost and Utilization Project. PARTICIPANTS: Pediatric (age ≤ 19 y) patients hospitalized for TBI (N=71,476) in the United States during 2001, 2004, 2007, and 2010. INTERVENTIONS: None. MAIN OUTCOME MEASURES: Primary outcome was proportion of patients discharged to rehabilitation after an acute care hospitalization among alive discharges. The secondary outcome was inpatient mortality. RESULTS: The relative risk of discharge to inpatient rehabilitation varied by as much as 3-fold among the states, and the relative risk of inpatient mortality varied by as much as nearly 2-fold. In the United States, approximately 1981 patients could be discharged to inpatient rehabilitation care if the observed variation in outcomes was eliminated. CONCLUSIONS: There was significant variation between states in both rehabilitation discharge and inpatient mortality after adjusting for variables known to affect each outcome. Future efforts should be focused on identifying the cause of this state-to-state variation, its relationship to patient outcome, and standardizing treatment across the United States.

Sociocultural and socioeconomic influences on type 2 diabetes risk in overweight/obese African-American and Latino-American children and adolescents.

PURPOSE: It is unclear whether sociocultural and socioeconomic factors are directly linked to type 2 diabetes risk in overweight/obese ethnic minority children and adolescents. This study examines the relationships between sociocultural orientation, household social position, and type 2 diabetes risk in overweight/obese African-American (n = 43) and Latino-American (n = 113) children and adolescents. METHODS: Sociocultural orientation was assessed using the Acculturation, Habits, and Interests Multicultural Scale for Adolescents (AHIMSA) questionnaire. Household social position was calculated using the Hollingshead Two-Factor Index of Social Position. Insulin sensitivity (SI), acute insulin response (AIRG) and disposition index (DI) were derived from a frequently sampled intravenous glucose tolerance test (FSIGT). The relationships between AHIMSA subscales (i.e., integration, assimilation, separation, and marginalization), household social position and FSIGT parameters were assessed using multiple linear regression. RESULTS: For African-Americans, integration (integrating their family's culture with those of mainstream white-American culture) was positively associated with AIRG (β = 0.27 ± 0.09, r = 0.48, P < 0.01) and DI (β = 0.28 ± 0.09, r = 0.55, P < 0.01). For Latino-Americans, household social position was inversely associated with AIRG (β = -0.010 ± 0.004, r = -0.19, P = 0.02) and DI (β = -20.44 ± 7.50, r = -0.27, P < 0.01). CONCLUSIONS: Sociocultural orientation and household social position play distinct and opposing roles in shaping type 2 diabetes risk in African-American and Latino-American children and adolescents.

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.