  Bayesian methods - either based on Bayes Factors or BIC - are now widely used
for model selection. One property that might reasonably be demanded of any
model selection method is that if a model ${M}_{1}$ is preferred to a model
${M}_{0}$, when these two models are expressed as members of one model class
$\mathbb{M}$, this preference is preserved when they are embedded in a
different class $\mathbb{M}'$. However, we illustrate in this paper that with
the usual implementation of these common Bayesian procedures this property does
not hold true even approximately. We therefore contend that to use these
methods it is first necessary for there to exist a "natural" embedding class.
We argue that in any context like the one illustrated in our running example of
Bayesian model selection of binary phylogenetic trees there is no such
embedding.
