  Occam's razor is a guiding principle that models should be simple enough to
describe observed data. While Bayesian model selection (BMS) embodies it by the
intrinsic regularization effect (IRE), how observed data scale the IRE has not
been fully understood. In the nonlinear regression with conditionally
independent observations, we show that the IRE is scaled by observations'
fineness, defined by the amount and quality of observed data. We introduce an
observable that quantifies the IRE, referred to as the Bayes specific heat,
inspired by the correspondence between statistical inference and statistical
physics. We derive its scaling relation to observations' fineness. We
demonstrate that the optimal model chosen by the BMS changes at critical values
of observations' fineness, accompanying the IRE's variation. The changes are
from choosing a coarse-grained model to a fine-grained one as observations'
fineness increases. Our findings expand an understanding of BMS's typicality
when observed data are insufficient.
