The novel framework for solving inverse problems in which all of my doctoral work has been grounded.
In the broadest terms possible, Data-Consistent Inversion (DCI) is a new approach to answer questions about the conditions under which observations can occur.
What makes it particularly special is that in a sense, this approach leverages the “best of both worlds” in the statistical inference world. The two schools of thought: Deterministic Optimization and Bayesian Statistics have different philosophical frameworks for answering questions pertaining to the explanation of observed data.
Bayesians update prior beliefs with new information through the use of likelihood functions constructed from observed data. How prior beliefs are chosen is a subject of contention since they influence conclusions.
Determinists attempt to explain the data by minimizing the misfit between predictions and observations. To prevent overfitting, regularization is usually employed to enforce certain structures in solutions, the strength of which is a modeling choice.
DCI updates initial descriptions of uncertainty only in directions informed by data. It is analogous to infinite regularization in the subspace orthogonal to the range of the operator. Initial descriptions play a similar role to the priors in Bayesian inversion, but are applied only to that subspace which is not described by the data.