Recursions for statistical multiple alignment

Algorithms are presented that allow the calculation of the probability of a set of sequences related by a binary tree that have evolved according to the Thorne-Kishino-Felsenstein model for a fixed set of parameters. The algorithms are based on a Markov chain generating sequences and their alignment at nodes in a tree. Depending on whether the complete realization of this Markov chain is decomposed into the first transition and the rest of the realization or the last transition and the first part of the realization, two kinds of recursions are obtained that are computationally similar but probabilistically different. The running time of the algorithms is O(Pi(i=1)(d) L-i), where L-i is the length of the ith observed sequences and d is the number of sequences. An alternative recursion is also formulated that uses only a Markov chain involving the inner nodes of a tree.