Irrational Exuberance: Correcting Bias in Probability Estimates

TL;DR

This paper introduces ECAP, a non-parametric empirical Bayes method that corrects bias in probability estimates, especially after selecting extreme events, improving accuracy in risk assessment.

Contribution

The paper develops ECAP, a flexible, non-parametric approach using Tweedie's formula to correct bias in probability estimates without restrictive assumptions.

Findings

ECAP improves probability estimates in simulations.

ECAP reduces bias in real-world data sets.

Theoretical analysis supports ECAP's effectiveness.

Abstract

We consider the common setting where one observes probability estimates for a large number of events, such as default risks for numerous bonds. Unfortunately, even with unbiased estimates, selecting events corresponding to the most extreme probabilities can result in systematically underestimating the true level of uncertainty. We develop an empirical Bayes approach "Excess Certainty Adjusted Probabilities" (ECAP), using a variant of Tweedie's formula, which updates probability estimates to correct for selection bias. ECAP is a flexible non-parametric method, which directly estimates the score function associated with the probability estimates, so it does not need to make any restrictive assumptions about the prior on the true probabilities. ECAP also works well in settings where the probability estimates are biased. We demonstrate through theoretical results, simulations, and an analysis of two real world data sets, that ECAP can provide significant improvements over the original probability estimates.