Characterization of basal pseudopod-like processes in ileal and colonic PYY cells.

The peptide tyrosine tyrosine (PYY) is produced and secreted from L cells of the gastrointestinal mucosa. To study the anatomy and function of PYY-secreting L cells, we developed a transgenic PYY-green fluorescent protein mouse model. PYY-containing cells exhibited green fluorescence under UV light and were immunoreactive to antibodies against PYY and GLP-1 (glucagon-like peptide-1, an incretin hormone also secreted by L cells). PYY-GFP cells from 15 μm thick sections were imaged using confocal laser scanning microscopy and three-dimensionally (3D) reconstructed. Results revealed unique details of the anatomical differences between ileal and colonic PYY-GFP cells. In ileal villi, the apical portion of PYY cells makes minimal contact with the lumen of the gut. Long pseudopod-like basal processes extend from these cells and form an interface between the mucosal epithelium and the lamina propria. Some basal processes are up to 50 μm in length. Multiple processes can be seen protruding from one cell and these often have a terminus resembling a synapse that appears to interact with neighboring cells. In colonic crypts, PYY-GFP cells adopt a spindle-like shape and weave in between epithelial cells, while maintaining contact with the lumen and lamina propria. In both tissues, cytoplasmic granules containing the hormones PYY and GLP-1 are confined to the base of the cell, often filling the basal process. The anatomical arrangement of these structures suggests a dual function as a dock for receptors to survey absorbed nutrients and as a launching platform for hormone secretion in a paracrine fashion.

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.

Fate of products of degradation processes: consequences for climatic change.

The end products of atmospheric degradation are not only CO2 and H2O but also sulfate and nitrate depending on the chemical composition of the substances which are subject to degradation processes. Atmospheric degradation has thus a direct influence on the radiative balance of the earth not only due to formation of greenhouse gases but also of aerosols. Aerosols of a diameter of 0.1 to 2 micrometer, reflect short wave sunlight very efficiently leading to a radiative forcing which is estimated to be about -0.8 watt per m2 by IPCC. Aerosols also influence the radiative balance by way of cloud formation. If more aerosols are present, clouds are formed with more and smaller droplets and these clouds have a higher albedo and are more stable compared to clouds with larger droplets. Not only sulfate, but also nitrate and polar organic compounds, formed as intermediates in degradation processes, contribute to this direct and indirect aerosol effect. Estimates for the Netherlands indicate a direct effect of -4 watt m-2 and an indirect effect of as large as -5 watt m-2. About one third is caused by sulfates, one third by nitrates and last third by polar organic compounds. This large radiative forcing is obviously non-uniform and depends on local conditions.

Interspecies avian brain chimeras reveal that large brain size differences are influenced by cell-interdependent processes.

Like humans, birds that exhibit vocal learning have relatively delayed telencephalon maturation, resulting in a disproportionately smaller brain prenatally but enlarged telencephalon in adulthood relative to vocal non-learning birds. To determine if this size difference results from evolutionary changes in cell-autonomous or cell-interdependent developmental processes, we transplanted telencephala from zebra finch donors (a vocal-learning species) into Japanese quail hosts (a vocal non-learning species) during the early neural tube stage (day 2 of incubation), and harvested the chimeras at later embryonic stages (between 9-12 days of incubation). The donor and host tissues fused well with each other, with known major fiber pathways connecting the zebra finch and quail parts of the brain. However, the overall sizes of chimeric finch telencephala were larger than non-transplanted finch telencephala at the same developmental stages, even though the proportional sizes of telencephalic subregions and fiber tracts were similar to normal finches. There were no significant changes in the size of chimeric quail host midbrains, even though they were innervated by the physically smaller zebra finch brain, including the smaller retinae of the finch eyes. Chimeric zebra finch telencephala had a decreased cell density relative to normal finches. However, cell nucleus size differences between each species were maintained as in normal birds. These results suggest that telencephalic size development is partially cell-interdependent, and that the mechanisms controlling the size of different brain regions may be functionally independent.