Estimating the probability of a rare event using imprecise probabilities (IP)/ probability bounds analysis (PBA)

If we must estimate the probability of a rare event, given little data and inadequate models, we are in deep water, to put it mildly. To make matters worse, in such instances, ordinary Bayesian inference relies heavily on model-parameter priors that are perhaps unrealistically precise. The method of imprecise probabilities (IP) or probability bounds analysis (PBA)¹ is a way of enforcing greater humility regarding the true state of our prior knowledge, and thus the range of feasible priors. Picard and Vander Wiel² describe an example based on high-explosive drop-test data, where the detonation of a high-explosive sample on which a heavy weight is dropped is the (undesirable) rare event. We reproduce their example using the GPMSA³ Bayesian inference code, and show how probability estimates depend on user choices regarding the allowable range of priors, the model form, and the search design in model-parameter space. We go on to consider the case of inertial-confinement fusion, where fusion ignition is the (desirable) rare (thus far) event.

1.     W. T. Tucker and S. Ferson, https://www.researchgate.net/publication/240192153_Probability_Bounds_Analysis_in_Environmental _Risk_Assessment (January 2003)

2.     R. R. Picard and S. A. Vander Wiel, LA-UR-16-23428 (May 2016)

3.     D. Higdon et al., JASA 103, 570-583 (2008)