Texas Engineers have developed a new smooth and continuous probability distribution system parameterized by percentile assessments gathered through expert elicitation, allowing decision analysts to model expert knowledge with greater fidelity and more easily than competing methods.

UT Austin Researchers: Christopher Hadlock, doctoral candidate in Operations Research & Industrial Engineering at The University of Texas at Austin, J. Eric Bickel, associate professor and director of the Graduate Program in Operations Research & Industrial Engineering at The University of Texas at Austin.

Discovery: A new smooth and continuous probability distribution system known as “J-QPD” parameterized by percentile assessments gathered through expert elicitation or data. This system is directly parameterized by percentile assessments (e.g., 10th, 50th, and 90th, percentiles), meaning that it always honors these percentiles, and that no optimization is required in assigning a distribution to these points – unlike existing methods.

Existing methods for assigning distributions to assessed percentiles often involve curve-fitting (e.g., least-squares), which generally involves nonlinear optimization. More importantly, however, fitted distributions almost never honor the assessed points. Alternative methods, such as discretization or maximum-entropy, have competitive tractability and honor the assessed points, but either neglect (discretization) or poorly represent (maximum-entropy) distribution tails.

Why It Matters: Capturing expert knowledge with high fidelity is a central part of the decision analysis process. Crudeness associated with methods such as curve-fitting, discretization, or maximum-entropy can yield poor and costly decisions in many business situations (e.g., to drill or not to drill for hydrocarbons), as demonstrated in the literature. In addition, the optimization required in curve-fitting is computationally inefficient compared to the J-QPD system, and some analysts may find these procedures slow or difficult to implement in practice.

How it Works: In decision analysis, analysts often elicit a symmetric percentile triplet (SPT) from an expert (e.g., 10th, 50th, 90th, or 5th, 50th, 95th percentiles). For many quantities, analysts also know something about support limits – for example, time and distance cannot be negative, and fractional quantities, such as market share assessments, must lie between zero and unity. Given this partial information, analysts often want a complete, smooth and continuous functional representation that honors the SPT and support limits.

By strategically applying quantile transformations and parameter manipulation to the highly-flexible Johnson SU system, the researchers produced the J-QPD system, which is parameterized by (and precisely honors any) any compatible combination of coherent SPT points and natural support limits via a simple and elegant expression. Beyond being quantile-parameterized, the J-QPD system is smooth, continuous, highly-flexible, and able to approximate a vast array of various commonly-named distributions (including normal, lognormal, beta, gamma, Weibull, Pareto) with potent accuracy – using one simple system. J-QPD also facilitates efficient Monte Carlo simulation via inverse transform sampling.

Published: The work will be published in the Decision Analysis journal in March, 2017: http://pubsonline.informs.org/page/deca/upcoming-issues

What's Next: The researchers are exploring improved methods of modeling dependence using quantile-parameterized distributions, as well as developing a spin-off distribution to the J-QPD system that is more flexible and more computationally efficient.