Marginal likelihoods, proper model evidence, and Bayes factors

Let $I=\{1,\dots ,N\}$ with $N\ge 2$ be a finite model index set. For each $i\in I$, let $({\mathrm{\Theta }}_i,{\mathcal{T}}_i)$ be a measurable parameter space, let ${\mathrm{\Pi }}_i$ be a probability measure (Probability measures and probability spaces) on it, and let $Q_i$ be a sampling kernel for a statistical experiment (Statistical experiments) on one common data space $(\mathcal{Y},\mathcal{A})$. Assume the experiments are dominated (Dominated statistical experiments) by the same supplied sigma-finite measure $\mu$. Fix a specified likelihood density (Likelihood densities) $f_i:{\mathrm{\Theta }}_i\times \mathcal{Y}\to [0,\mathrm{\infty }]$ such that $Q_i(\theta ,B)={\int }_B{f}_i(\theta ,y)\mu (dy)(B\in \mathcal{A})\mathrm{.}$ Require that each $f_i$ is jointly measurable in parameter and data; separate likelihood versions do not imply this joint measurability. Assign prior model weights $w_i>0$ with ${\sum }_i{w}_i=1$. The marginal likelihood, or model evidence, at $y$ is the possibly extended value $m_i(y):={\int }_{{\mathrm{\Theta }}_i}{f}_i(\theta ,y){\mathrm{\Pi }}_i(d\theta )\in [0,\mathrm{\infty }]\mathrm{.}$ This is the BCS-02 evidence normalizer (Evidence normalizer and fiberwise posterior update) specialized to a proper model prior, and is the prior-predictive density relative to the specified common data measure. For models $i,j$ whose evidences at $y$ are both finite and positive, the Bayes factor for $j$ against $i$ is $B{F}_{ji}(y):=\frac{m_j(y)}{m_i(y)}\mathrm{.}$ The corresponding prior model odds are $w_j\mathrm{/}{w}_i$. No Bayes factor is defined here using an arbitrarily scaled improper parameter prior.

A Bayes factor is an odds multiplier. This page supplies no universal cutoff that converts its numerical magnitude into a binary decision about which model is true.