  The Bayes factor is the gold-standard figure of merit for comparing fits of
models to data, for hypothesis selection and parameter estimation. However it
is little used because it is computationally very intensive. Here it is shown
how Bayes factors can be calculated accurately and easily, so that any
least-squares or maximum-likelihood fits may be routinely followed by the
calculation of Bayes factors to guide the best choice of model and hence the
best estimations of parameters. Approximations to the Bayes factor, such as the
Bayesian Information Criterion (BIC), are increasingly used. Occam's razor
expresses a primary intuition, that parameters should not be multiplied
unnecessarily, and that is quantified by the BIC. The Bayes factor quantifies
two further intuitions. Models with physically-meaningful parameters are
preferable to models with physically-meaningless parameters. Models that could
fail to fit the data, yet which do fit, are preferable to models which span the
data space and are therefore guaranteed to fit the data. The outcomes of using
Bayes factors are often very different from traditional statistics tests and
from the BIC. Three examples are given. In two of these examples, the easy
calculation of the Bayes factor is exact. The third example illustrates the
rare conditions under which it has some error and shows how to diagnose and
correct the error.
