Análisis de sensibilidad bayesiana a través de clases de distribuciones a priori: teoría y aplicaciones

This Ph.D. dissertation provides contributions in the study of robustness in decision-making problems from a Bayesian point of view. We bring interesting results related with robust Bayesian analysis which make this studies easier. Then, these results are applied in the study of real problems in different contexts: actuarial or financial risk, metrology and reliability theory. Consequently, the thesis is divided into three chapters, where each of them are involved in these different backgrounds. Roughly speaking, Bayesian Statistics obtain the posterior distribution of an underlying univariate or multivariate parameter as a consequence of the likelihood function from an initial sample and a prior information of the parameter according to the Bayes' rule. So, the main interest of the Bayesian point of view is the contribution not only the initial sample that we have but also introducing more information stemming from some experts. That prior information may come in different forms, although it is usually based on the experts' knowledge, which will give enough information to make decisions. In Bayesian inference is fundamental a high precision in the decision maker's judgement, specially regarding his beliefs and preferences. In the Bayesian decision framework, the prior distribution is determined in the set of states given by the experts and it is used to obtain a posterior distribution and a posterior quantity of interest depending on the problem. Usually, that quantity is such that minimizes the expected loss, which is known as the Bayes action, especially in the univariate case. It is common in the Bayesian decision framework starting from a unique prior distribution. However, there are plenty of criticism on that issue: has been well selected the prior distribution from the prior knowledge? Has been introduced any biased information, i.e., there exist any subjectivity in the specific prior distribution? How difficult is to express mathematically the experts' prior knowledge? So, the problem gets more complicated when there exist inaccuracies in the choice of the prior distribution. Therefore, a Bayesian robust analysis seems to be essential. The main goal of Bayesian robustness is to quantify and interpret the uncertainty induced by partial knowledge of one (or more) of the three elements in the analysis. Those three elements are the prior distribution, the loss function and the likelihood function. However, thorough this work, we will mainly focus on prior uncertainty for two major reasons. First, use of priors have been criticized by detractors of the Bayesian approach and Bayesian robustness provides a way to address such issue. Second, there is a practical difficulty in specifying exactly a prior corresponding to the experts' knowledge. Bayesian analysis in complex problems typically entails messy computations, and most times one cannot afford the additional computational burden that would be imposed by a formal robustness analysis. Particularly when one try to compute the range of a posterior quantity of interest. So, we can find many papers in the literature where authors try to simplify that procedures. In particular, this work has focused on it. On the other hand, the Markov Chain - Monte Carlo (MCMC) algorithms appears as an essential tool in the Bayesian decision problem. We refer to the enormous impact that MCMC methods have had on Bayesian analysis, and taking into account that Bayesian robustness methodology will need to be compatible with MCMC methods to become widely used. Though much additional work needs to obtain good results of the robust Bayesian techniques by using MCMC, it is the best way to obtain quality samples and values for the quantity of interest. Then, this work is focused on replacing a single prior distribution by a class of priors but developing classes of prior distribution which make easier the computation of the classes of posterior distribution and, therefore, the computation of the set of the quantity of interest. In order to carry out the Bayesian sensitivity analysis, it will be important to define some useful tools: the univariate and multivariate stochastic orders and the distortion functions. First, stochastic orderings are specially important, providing information about how two distributions can be compared depending on what we are looking for. For example, the simplest one is by considering classical characteristics of the distributions, as the mean value or the standard deviation. However, these comparisons are not good enough because we summarize all information in just a single measure. In this way, stochastic orders represent a powerful tool which allows us to compare two random variables in terms of different criteria. Here we list some of the stochastic orders which we will use along this dissertation: the univariate and multivariate usual stochastic order, the increasing convex (concave) order, the univariate and multivariate likelihood ratio order and the uniform conditional variability order. We will see their formal definitions and some interesting properties, including the chain of implication among all of them. In addition, distortion functions play an important role not only in this dissertation but also in different fields that appear on it, as in actuarial theory. A distortion function h is a non-decreasing continuous function such that h(0)=0 and h(1)=1. For each distortion, we can find a distorted distribution function. This idea will be explained better in the introduction, besides more interesting properties of it. It is worth mentioning that in the multivariate case there exist different ways to define the distortion functions. So, rather than choice the more suitable option every time, we will use a natural extension given by the concept of weight functions and weighted densities in the multivariate case. We provide its definition and the main properties. To summarize, this PhD dissertation will focus on develop new ways to study Bayesian robustness in the prior distribution using different tools as the classical ones. Among all these tools, we will use stochastic orders, distortion functions and weight functions. It will be considered as the starting point of this work a new class of prior distribution that they show: the Distorted Band. We will find more information about this class along the dissertation. Also, this work gives several examples and ideas to indicate the importance and uses of robustness in a Bayesian setting. The main idea is to develop new results that allow us to make sensitivity analysis in different fields of application: actuarial risk, metrology and reliability theory. Finally, the key idea is to obtain a new multivariate class of prior distribution likewise the Distorted Band

Structure and Evolution of Human Immunoglobulin Cγ Genes

In order to learn about the evolution of the human immunoglobulin Cγ gene family, the structural features of individual Cγ genes were examined. The complete nucleotide sequences were determined for three members of the gene family-the Cγ1, Cγ2, and Cγ4 genes. A comparison of these sequences with those of the three reported mouse Cγ genes (Cγ1, Cγ2a, Cγ2b) fails to reveal any pairs of corresponding genes in the two species. Moreover, the sequence homology shared by human Cγ genes in both coding and noncoding regions (about 95%) is significantly greater than that seen within the mouse Cγ family (about 70-80%). The presumably neutral mutations accumulated in the noncoding regions of the human genes have been used to estimate that approximately 6-8 million years have elapsed since the divergence of these genes from a common ancestral sequence. This divergence is considerably more recent than inferred for the mouse Cγ genes, and suggests that gene duplication or gene correction events have occurred more recently in humans than in mice. In contrast to the CH domain exons and adjacent noncoding regions, the hinge exons of human Cγ genes are quite divergent both in length and sequence. This coding sequence variability is seen to extend into the regions of CH domains which border the hinge in the polypeptide chain. This divergence is interpreted as being the result of natural selection for particular hinge structures in the IgG subclasses. The implication is that these polypeptide regions are important for immunologic effector functions carried out by IgG molecules. The arrangement of the Cγ2 and Cγ4 genes in human chromosomal DNA has been determined to be 5'-Cγ2-17 kilobase pairs-Cγ4-3'. The genetic processes generating hybrid IgG molecules from these two genes are discussed, along with the relationship of these processes to gene duplication and gene correction.</p

The Origins, Specificity, and Potential Biological Relevance of Human Anti-IgG Hinge Autoantibodies

Human anti-IgG hinge (HAH) autoantibodies constitute a class of immunoglobulins that recognize cryptic epitopes in the hinge region of antibodies exposed after proteolytic cleavage, but do not bind to the intact IgG counterpart. Detailed molecular characterizations of HAH autoantibodies suggest that they are, in some cases, distinct from natural autoantibodies that arise independent of antigenic challenge. Multiple studies have attempted to define the specificity of HAH autoantibodies, which were originally detected as binding to fragments possessing C-terminal amino acid residues exposed in either the upper or lower hinge regions of IgGs. Numerous investigators have provided information on the isotype profiles of the HAH autoantibodies, as well as correlations among protease cleavage patterns and HAH autoantibody reactivity. Several biological functions have been attributed to HAH autoantibodies, ranging from house-cleaning functions to an immunosuppressive role to restoring function to cleaved IgGs. In this review, we discuss both the historic and current literature regarding HAH autoantibodies in terms of their origins, specificity, and proposed biological relevance