Probabilistic Path Hamiltonian Monte Carlo

6. Appendix 6.1. Properties of the phylogenetic posterior distribution Assumption 2.3 for the phylogenetic posterior distribution. Recall that L(⌧, q) denotes the likelihood function of the tree T = (⌧, q), we have U(⌧, q) = logL(⌧, q) log ⇡0(⌧, q) Since log ⇡0(⌧, q) is assumed to satisfy the Assumption 2.3, we just need to prove that the phylogenetic likelihood function is smooth while each orthant and is continuous on the whole space. Without loss of generality, we consider the case when a single branch length of some edge e is contracted to zero. To investigate the changes in the likelihood function and its derivatives, we first fix all other branches, partition the set of all extensions of according to their labels at the end points of e, and split E(T ) into two sets of edges Eleft and Eright corresponding to the location of the edges with respect to e. The likelihood function of the tree T = (⌧, q) can be rewritten as L(T ) = S Y s=1 X