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We analytically derive the first and second derivatives of the likelihood in maximum likelihood methods for phylogeny. These results enable the Newton-Raphson method to be used for maximising likelihood, which is important because there is a need for faster methods for optimisation of parameters in maximum likelihood methods. Furthermore,  the calculation of the Hessian matrix also opens up possibilities for standard likelihood theory to be applied, for inference in phylogeny and for model selection problems.  Another application of the Hessian matrix is local influence analysis, which can be used for detecting a number of biologically interesting phenomena. The pruning algorithm has been used to speed up computation of likelihoods for a tree. We explain how it can be used to speed up the computation for the first and second derivatives of the likelihood with respect to branch lengths and other parameters. The results in this paper apply not only to bifurcating trees, but also to general multifurcating trees. We demonstrate the use of our Hessian calculation for the three applications listed above, and compare with existing methods for those applications.

Hessian Calculation for Phylogenetic Likelihood based on the Pruning Algorithm and its Applications