Use of ABIC and Invention of Inversion Methods

Abstract:

Bayesian inference is a powerful tool in inversion analyses of geophysical problems, because observed data are commonly inaccurate and insufficient in these problems. In Bayesian inference, we always encounter a problem in determining the relative weight between observed data and prior information. ABIC (Akaike's Bayesian Information Criterion) gives a useful solution to this problem particularly for linear inverse problems, by maximizing the marginal likelihood for the relative weight. In general, we subjectively construct a Bayesian model, which consists of a family of parametric models with different values of the relative weight giving different parametric models; ABIC enables us to objectively select a specific model among the parametric models. In principle, ABIC gives us an inverse solution that mostly follows observed data when we have enough amount of data with good accuracy, and gives us an inverse solution that mostly follows prior information when observed data are insufficient and/or inaccurate (see the attached image).

In inversion analyses using ABIC, we do not manually adjust the relative weight. Hence, we quite easily obtain geophysically unrealistic results. Because of that, someone may think that inversion analyses using ABIC is difficult in dealing with or even unreliable. However, this characteristic is an excellent point of ABIC. If we obtain a geophysically unrealistic result, this implies that some problems are hidden in the inversion method.

In this talk, we show an example of the invention of inversion methods inspired by ABIC: the importance of covariance components including modeling errors. As shown by this example, we can get closer to the true solution not by manually adjusting the relative weight to obtain a seemingly good-looking result, but by determining the relative weight statistically. It is a harder way to determine the relative weight statistically, but we should pursue this way to understand geophysical problems more appropriately.