Approximations, simulation, and accuracy of multivariate discrete probability distributions in decision analysis
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Many important decisions must be made without full information. For example, a woman may need to make a treatment decision regarding breast cancer without full knowledge of important uncertainties, such as how well she might respond to treatment. In the financial domain, in the wake of the housing crisis, the government may need to monitor the credit market and decide whether to intervene. A key input in this case would be a model to describe the chance that one person (or company) will default given that others have defaulted. However, such a model requires addressing the lack of knowledge regarding the correlation between groups or individuals. How to model and make decisions in cases where only partial information is available is a significant challenge. In the past, researchers have made arbitrary assumptions regarding the missing information. In this research, we developed a modeling procedure that can be used to analyze many possible scenarios subject to strict conditions. Specifically, we developed a new Monte Carlo simulation procedure to create a collection of joint probability distributions, all of which match whatever information we have. Using this collection of distributions, we analyzed the accuracy of different approximations such as maximum entropy or copula-models. In addition, we proposed several new approximations that outperform previous methods. The objective of this research is four-fold. First, provide a new framework for approximation models. In particular, we presented four new models to approximate joint probability distributions based on geometric attributes and compared their performance to existing methods. Second, develop a new joint distribution simulation procedure (JDSIM) to sample joint distributions from the set of all possible distributions that match available information. This procedure can then be applied to different scenarios to analyze the sensitivity of a decision or to test the accuracy of an approximation method. Third, test the accuracy of seven approximation methods under a variety of circumstances. Specifically, we addressed the following questions within the context of multivariate discrete distributions: Are there new approximations that should be considered? Which approximation is the most accurate, according to different measures? How accurate are the approximations as the number of random variables increases? How accurate are they as we change the underlying dependence structure? How does accuracy improve as we add lower-order assessments? What are the implications of these findings for decision analysis practice and research? While the above questions are easy to pose, they are challenging to answer. For Decision Analysis, the answers open a new avenue to address partial information, which bing us to the last contribution. Fourth, propose a new approach to decision making with partial information. The exploration of old and new approximations and the capability of creating large collections of joint distributions that match expert assessments provide new tools that extend the field of decision analysis. In particular, we presented two sample cases that illustrate the scope of this work and its impact on uncertain decision making.