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CALIBRATING THE CLOCK: USING STOCHASTIC PROCESSES TO MEASURE THE RATE OF EVOLUTION 130 to construct true maximum likelihood estimators of the parameter θ. It can also be used to compare "likelihoods" of the different ancestral labelings. The corresponding theory of the unrooted genealogical trees that arise when the ancestral labeling is unknown has recently been developed by Griffiths and Tavaré (1994c), and this leads to a practical computational method for estimating θ by maximum likelihood. Our analysis of the mitochondrial data set has shown that while parts of the region are consistent with a simple evolutionary model, there are sites which are behaving in a more complicated way. In the next section, we describe a finitely-many-sites model that is useful for modeling regions in which back mutations have occurred. K-Allele Models We turn first to the K-allele model. In this process, we assume that there are K possible alleles at the locus in question. When a mutation occurs to an allele of type i, there is a probability mij that the resulting allele is of type j. To allow for different rates of substitution for different alleles, we can have mii > 0, and we write M = (mij). The effects of mutation along a given line are now modeled by a continuous time Markov chain whose transition matrix P(t) ⡠(pij(t)) gives the probabilities that a gene of type i has been replaced by a descendant gene of type j a time t later. Indeed, where I is the K à K identity matrix, so that the generator of the mutation process is (5.13)
CALIBRATING THE CLOCK: USING STOCHASTIC PROCESSES TO MEASURE THE RATE OF EVOLUTION 131 It is worth pointing out that a given Q matrix can be represented in more than one way in the form (5.13), so that θ, for example, is not identifiable without further assumptions. However, the rates qij, j â i are identifiable. If Q has a stationary distribution Ï = (Ï1, Ï2,. . ., ÏK) satisfying Ï Q = 0, , and if the common ancestor of the sample has distribution Ï, then the distribution of a gene at any point in the tree is also Ï, and the process is stationary. From the data analyst's perspective, the sample of n genes can be sorted into a vector N â¡ (N1, N2,. . .,NK) of counts, there being Nj alleles of type j in the sample. Surprisingly, the stationary distribution of N is known explicitly only for the special case This is equivalent to the independent mutations case in which θ = ε1 + ε2 +. . .+ εK,Ïi = εi / θ, and mij = Ïj for all i and j. In this case, Wright's Formula (Wright, 1968) can be used to show that (5.14) for n = (n1,n2,. . .,nK), nj ⥠0 for j = 1,2,. . .,K, and n = n1 + n2 +. . .+nK. In the next section, we use this result for the case K = 2 . If (5.15) then equation (5.14) specializes to (5.16)
