Statistical Measures of Uncertainty in Inverse Problems

Abstract

References & Acknowledgements

Outline

Inverse Problems as Statistics

Models

Forward Problems in Geophysics

Inverse Problems

Geophysical Inverse Problems

Elements of the Statistical View

Deterministic and Statistical Perspectives: Connections

More Notation

Linear Forward Problems

Linear Forward Problems, contd.

Linear Inverse Problems

Ex: Sampling w/ systematic and random error

Sketch: Identifiability

Backus-Gilbert Theory

Sketch: Backus-Gilbert

Backus-Gilbert⁺⁺: Necessary conditions

Backus-Gilbert⁺⁺: Sufficient Conditions

Decision Rules

Estimators

Comparing Decision Rules

Loss and Risk

Strategy

Minimax is Bayes for least favorable prior

Common Risk: Mean Distance Error (MDE)

Bias

Sketch: Regularization

Minimax Estimation of Linear parameters:
Hilbert Space, Gaussian error

Modulus of Continuity

Distinguishing two models

L₁ and Hellinger distances

Consistency in Linear Inverse Problems

Importance of the Error Distribution

Summary