  In statistics, there are a variety of methods for performing model selection
that all stem from slightly different paradigms of statistical inference. The
reasons for choosing one particular method over another seem to be based
entirely on philosophical preferences. In the case of non-nested model
selection, two of the prevailing techniques are the Bayes Factor and the
likelihood ratio. This article focuses on reconciling the likelihood ratio and
the Bayes Factor for comparing a pair of non-nested models under two different
problem frameworks typical in forensic science, the common-source and the
specific-source problem. We show that the Bayes Factor can be expressed as the
expected value of the corresponding likelihood ratio function with respect to
the posterior distribution for the parameters given the entire set of data
where the set(s) of unknown-source observations has been generated according to
the second model. This expression leads to a number of useful theoretic and
practical results relating the two statistical approaches. This relationship is
quite meaningful in many scientific applications where there is a general
confusion between the various statistical methods, and particularly in the case
of forensic science.
