  In many hypothesis testing applications, we have mixed priors, with
well-motivated informative priors for some parameters but not for others. The
Bayesian methodology uses the Bayes factor and is helpful for the informative
priors, as it incorporates Occam's razor via multiplicity or trials factor in
the Look Elsewhere Effect. However, if the prior is not known completely, the
frequentist hypothesis test via the false positive rate is a better approach,
as it is less sensitive to the prior choice. We argue that when only partial
prior information is available, it is best to combine the two methodologies by
using the Bayes factor as a test statistic in the frequentist analysis. We show
that the standard frequentist likelihood-ratio test statistic corresponds to
the Bayes factor with a non-informative Jeffrey's prior. We also show that
mixed priors increase the statistical power in frequentist analyses over the
likelihood ratio test statistic. We develop an analytic formalism that does not
require expensive simulations using a statistical mechanics approach to
hypothesis testing in Bayesian and frequentist statistics. We introduce the
counting of states in a continuous parameter space using the uncertainty volume
as the quantum of the state. We show that both the p-value and Bayes factor can
be expressed as energy versus entropy competition. We present analytic
expressions that generalize Wilks' theorem beyond its usual regime of validity
and work in a non-asymptotic regime. In specific limits, the formalism
reproduces existing expressions, such as the p-value of linear models and
periodograms. We apply the formalism to an example of exoplanet transits, where
multiplicity can be more than $10^7$. We show that our analytic expressions
reproduce the p-values derived from numerical simulations.
