Approximating multivariate distributions with cumulative residual entropy: a study on dynamic integrated climate-economy model

Abstract

The complexity of real world decision problems is exacerbated by the need to make decisions with only partial information. How to model and make decisions in situations where only partial preference information is available is a significant challenge in decision analysis practice. In most of the studies, the probability distributions are approximated by using the mass function or density function of the decision maker. In this dissertation, our aim is to approximate representative probability and utility functions by using cumulative distribution functions instead of density/mass functions. This dissertation consists of four main sections. The first two sections introduce the proposed methods based on cumulative residual entropy, the third section compares the proposed approximation methods with the methods in information theory literature, and the final section of the dissertation discusses the cumulative impact of integrating uncertainty into the DICE model. In the first section of the dissertation, we approximate discrete joint probability distributions using first-order dependence trees as well as the recent concept of cumulative residual entropy. We formulate the cumulative residual Kullback-Leibler (KL)-divergence and the cumulative residual mutual information measures in terms of the survival function. We then show that the optimal first-order dependence tree approximation of the joint distribution using the cumulative Kullback-Leibler divergence is the one with the largest sum of cumulative residual mutual information pairs. In the second part of the dissertation, we approximate multivariate probability distributions with cumulative probability distributions rather than density functions in maximum entropy formulation. We use the discrete form of maximum cumulative residual entropy to approximate joint probability distributions to elicit multivariate probability distributions using their lower order assessments. In the third part of the dissertation, we compare several approximation methods to test the accuracy of different approximations of joint distributions with respect to the true distribution from the set of all possible distributions that match the available information. A number of methods have beeb presented in the literature for joint probability distribution approximations and we specifically compare those approximation methods that use information theory to approximate multivariate probability distributions. Finally, we study whether uncertainty significantly affects decision making especially in global warming policy decisions and integrate climatic and economic uncertainties into the DICE model to ascertain the cumulative impact of integrating uncertainty on climate change by applying cumulative residual entropy into the DICE model