Texas Engineers have developed a new guidance for practitioners who use discrete approximations (discretizations) to continuous probability distributions, by showing how errors in probability distributions themselves compare to various approximation methods.
UT Austin Researchers: Robert Hammond, Decision Analyst at Chevron and former doctoral student in the Graduate Program in Operations Research & Industrial Engineering at The University of Texas at Austin. Eric Bickel, associate professor and director of the Graduate Program in Operations Research & Industrial Engineering at The University of Texas at Austin.
Discovery: Choosing a discretization method for use in analysis is an important decision even when the underlying continuous probability distribution is not exact. Additionally, the specific attributes of the distribution (e.g., the mean or variance) that the analyst seeks to preserve are important factors in determining the appropriate discretization method.
Why It Matters: Decision analysis practitioners often discretize continuous probability distributions for computational tractability and to facilitate probability assessments. Several previous research studies showed large differences in the accuracies of various discretization methods, however, practitioners commonly have a preferred method that they apply in most or all situations. Errors in probability assessments are often assumed to be so much larger than discretization errors that the choice of discretization method is not important. This research shows that the choice of method is in fact important for many cases of the assessment error levels seen in practice.
How it Works: The process of assessing a probability distribution for an uncertain quantity has assessment error, meaning that it might not precisely represent the actual uncertainty. This distribution is then discretized, which introduces discretization error. Different discretization methods produce different approximations, and the difference in these approximations is called the discretization precision. Higher precision between discretization methods means that the choice of method has less impact.
The researchers compare discretization precision to assessment error using several beta distributions, covering a wide variety of distribution shapes. Each of these distributions in turn is taken as the assessed distribution, with error ranges around the 10th, 50th, and 90th percentile assessments. These define a set of potential true distributions that are feasible for the given percentile assessments and error range. The differences in moments of the assessed distribution and the potential true distributions are compared to the discretization precision between pairs of discretization methods.
The research found that if the analyst seeks only to preserve the mean of the beta distributions, then discretization precision is usually high relative to assessment error, and discretization choice has little impact. However, if the analyst also seeks to preserve the variance, or higher moments, of the distributions, then discretization choice is important even for high levels of assessment error.
Published: Decision Analysis, “Discretization Precision and Assessment Error”
What's Next: Avenues for future work include expanding this work to other types of distributions (e.g., lognormal), examining the question of discretization importance in the context of specific decision problems, and examining the compounding effects of multiple discretizations and multivariate discretization.
