Análisis de sensibilidad bayesiana a través de clases de distribuciones a prioriteoría y aplicaciones

Resumen

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.