  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.
