Akaike's Bayesian information criterion (ABIC) or not ABIC for geophysical inversion

TL;DR

This paper critically examines the statistical foundations of ABIC in geophysical inversion, revealing its limitations in estimating measurement and prior variances, especially when noise levels are unknown.

Contribution

It provides a new derivation of ABIC's statistical basis and highlights its biases and limitations in practical geophysical inverse problems.

Findings

ABIC estimates measurement variance by maximizing the marginal distribution.

ABIC's regularization parameter relates to the ratio of measurement and prior variances.

When noise level is unknown, ABIC tends to produce biased variance estimates.

Abstract

Akaike's Bayesian information criterion (ABIC) has been widely used in geophysical inversion and beyond. However, little has been done to investigate its statistical aspects. We present an alternative derivation of the marginal distribution of measurements, whose maximization directly leads to the invention of ABIC by Akaike. We show that ABIC is to statistically estimate the variance of measurements and the prior variance by maximizing the marginal distribution of measurements. The determination of the regularization parameter on the basis of ABIC is actually equivalent to estimating the relative weighting factor between the variance of measurements and the prior variance for geophysical inverse problems. We show that if the noise level of measurements is unknown, ABIC tends to produce a substantially biased estimate of the variance of measurements. In particular, since the prior mean is generally unknown but arbitrarily treated as zero in geophysical inversion, ABIC does not produce a reasonable estimate for the prior variance either.