The inverse problem of determining information about the state of a physical system from observations of its behavior is fundamental to scientific inference and engineering design. Frequently, this can be formulated as computing a probability measure on physical characteristics of a system from observed data on the output of a model of system behavior. In abstract terms, this is the empirical stochastic inverse problem for a random vector on a probability space with an unknown probability measure. Over the last fifteen years, collaborators and I have developed a general Bayesian approach to the formulation and solution of this problem. Our approach has a solid theoretical foundation that avoids alterations of the model like regularization as well as unrealistic and limiting assumptions about prior knowledge of system characteristics, allows for numerical solution by a novel importance sampling approach, and provides a platform to address critical issues arising in the practical application to scientific and engineering problems. I will lay out the theoretical and computational foundation of our approach with the details motivated by practical applications including optimizing car mileage, cooking steaks, hurricane storm surge forecasting, and forecasting COVID surges. Time permitting, I will discuss the relationship with common Bayesian statistics.

Refreshments will be served preceding the talk, starting at 2:45.

We gratefully acknowledge funding support from the Pacific Institute of Mathematical Sciences (PIMS) and organizational support from SCAIM.