# Quantile-parameterized methods for quantifying uncertainty in decision analysis

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In decision analysis, analysts must often encode an expert’s uncertainty of a quantity (e.g., the volume of oil reserves in a proven, but undeveloped oil field) using a probability distribution. This is commonly done by eliciting a triplet of low-base-high percentile assessments, such as the {10th, 50th, 90th} percentiles, from the expert, and then fitting a probability distribution from a well-known family (e.g., lognormal) to the assessed quantile-probability (or QP) pairs. However, curve fitting often requires non-linear, non-convex optimization over a distribution parameter space, and the fitted distribution often never honors the assessed QP pairs – reducing both the fidelity of the model, and trust in the analysis. The development of quantile-parameterized distributions (or QPDs), distributions that are parameterized by, and thus precisely honor the assessed QP pairs, is a very important yet nascent topic in decision analysis, and contributions in the literature are sparse. This dissertation extends existing work on QPDs by strategically developing a new smooth probability distribution system (known as J-QPD) that is parameterized by (and honors) assessed QP pairs. J-QPD also honors various natural support regimes – for example: bounded (e.g., fractional uncertainties, such as market shares, are necessarily bounded between zero and one); semi-bounded (e.g., volume of oil reserves is necessarily non-negative, but may have no well-defined upper bound); etc. We then show that J-QPD is maximally-feasible, highly flexible, and approximates the shapes of a vast array of commonly-named distributions (e.g., normal, lognormal, beta, etc.) with potent accuracy, using a single system. This work also presents efficient, high-fidelity methods for capturing dependence between two or more uncertainties by combining J-QPD with modern correlation assessment and modeling techniques. We then provide an application of J-QPD to a famous decision analysis example, demonstrating how J-QPD facilitates rapid Monte Carlo simulation, and how its implementation can aid actual decisions that might otherwise be made wrongly if commonly-used discrete methods are used. We conclude by noting important tradeoffs between J-QPD and existing QPD systems, and offer several extensions for future research, including a first look at designing new discrete distributions using J-QPD.