MCMC sampling of tree structure

I want to do Metropolis Hastings sampling of a tree structure $\mathcal{T}$ (in the space where the tree size is fixed) from a posterior distribution:

$p(\mathcal{T}|X) \propto p(X|\mathcal{T})p(\mathcal{T})$ $\qquad \qquad$ (1)

where $X$ is observed data. Suppose I can calculate the RHS easily, then the only thing I need is a proposal function. What proposal functions do people use in such kind of tree structure sampling?

• Could you describe the "tree structure" in greater detail? It can be understood in different ways and so lead to misleading answers.
– Tim
Jun 16, 2016 at 7:45
• @Tim It's just typical tree in graph theory. You can understand it in an evolution context: the tree nodes are species. Putting the nodes together as tree in different ways give you different likelihood.
– Lii
Jun 16, 2016 at 17:23
• Maybe it's just not my area of expertise, but how do you define "likelihood of a tree"?
– Tim
Jun 16, 2016 at 18:32
• It's not likelihood of tree, but likelihood of observed data (the $p(X|\mathcal{T})$ part in equation (1) ).
– Lii
Jun 16, 2016 at 19:25
• But you say that different trees lead to different likelihoods so from what you are saying it sounds like it is dependent of the tree..? As you can see, it is not totally clear what do you mean...
– Tim
Jun 16, 2016 at 19:30