A Bifocal Measure of Expected Ambiguity in Bayesian Nonlinear Parameter Estimation

Abstract

We present a novel approach to define and calculate the expected uncertainty of Bayesian parameter estimates, prior to collecting any observational data. This can be used to design investigation techniques or experiments that minimize expected uncertainty. Our approach accounts fully for nonlinearity in the parameter–observation relationship, which is neither the case for the Bayesian D- and A-optimality criteria most commonly used in experimental design, nor the case for most other derivative- or information matrix-based experimental design techniques. Our method is based on analyzing pairs of parameter estimates, thus forming a “bifocal” measure of ambiguity. Derivatives of observable data with respect to parameter values are neither required nor calculated. For linear models, our new measure is equivalent to expected posterior variance, and it is closely related to expected posterior variance in nonlinear models.

Keywords:

• Bayesian methods
• Decision theory
• Frequency estimation
• Microseismic location
• Nonlinear models
• Optimal design.

ACKNOWLEDGMENTS

We thank the referees and, in particular, the Associate Editor and the Editor, whose comments and suggestions have been very helpful in clarifying this presentation.