in question.

Note also that the probability of a gene tree on a subset of the taxa can be obtained by summing probabilities of gene trees on the full set that display the given gene tree when restricted to the subset. Such marginalization of a gene tree distribution to fewer taxa is possible for either rooted or unrooted gene trees. Consequently, for most arguments in [@adr2011] it was sufficient to focus on small trees, with at most five taxa. Indeed, similar marginalization arguments are standard throughout phylogenetic theory.

Unfortunately, a marginalization approach fails for studying clades when the species tree is unknown. Given clade probabilities arising from an $n$-taxon species tree $\sigma$ under the multispecies coalescent, one would like to be able to determine clade probabilities arising from an induced $k$-taxon tree displayed on $\sigma$. However, probabilities of clades on the $k$-taxon induced tree cannot be obtained from a linear combination of the clade probabilities associated with the $n$-taxon species tree in a way that is independent of the species tree topology. We demonstrate this formally is the case where $k=3$ and $n=4$ in Appendix \[ap:linear\].

This inability to marginalize clade probabilities without knowing the species tree topology motivated looking 