  Building on recent work in statistical science, the paper presents a theory
for modelling natural phenomena that unifies physical and statistical paradigms
based on the underlying principle that a model must be nondimensionalizable.
After all, such phenomena cannot depend on how the experimenter chooses to
assess them. Yet the model itself must be comprised of quantities that can be
determined theoretically or empirically. Hence, the underlying principle
requires that the model represents these natural processes correctly no matter
what scales and units of measurement are selected. This goal was realized for
physical modelling through the celebrated theories of Buckingham and Bridgman
and for statistical modellers through the invariance principle of Hunt and
Stein. Building on recent research in statistical science, the paper shows how
the latter can embrace and extend the former. The invariance principle is
extended to encompass the Bayesian paradigm, thereby enabling an assessment of
model uncertainty. The paper covers topics not ordinarily seen in statistical
science regarding dimensions, scales, and units of quantities in statistical
modelling. It shows the special difficulties that can arise when models involve
transcendental functions, such as the logarithm which is used e.g. in
likelihood analysis and is a singularity in the family of Box-Cox family of
transformations. Further, it demonstrates the importance of the scale of
measurement, in particular how differently modellers must handle ratio- and
interval-scales
