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ijcrb.webs.com INTERDISCIPLINARY JOURNAL OF CONTEMPORARY RESEARCH IN BUSINESS In Figure. 4. Transition probabilities. The hidden states of the HMM represent different tree topologies, and state transitions correspond to recombination events. The transition probability v is the probability that on moving from a site in the DNA sequence alignment to an adjacent site, no topology change occurs. If a topology change does occur, we assume that, a priori, all transitions are equally likely The parameters w, θ, and v need to be estimated. 3. Heuristic Parameter Estimation: HMM-Heuristic McGuire, Wright, and Prentice (2000) estimated the branch lengths w for each tree topology separately with maximum likelihood. This approach is suboptimal. For a proper estimation of the branch lengths of a recombinant tree, one would have to restrict the parameter estimation to the recombinant region. The location of this region, however, is not known in advance. Estimating the branch lengths from the whole DNA sequence alignment leads to seriously distorted values, as demonstrated by Husmeier and Wright (2001) because the estimation includes regions of the alignment for which the tree topology is incorrect. A heuristic way to address this problem, suggested by McGuire, Wright, and Prentice (2000), is to estimate the branch lengths from a subregion of the alignment. The length of this region should be matched to the length of the recombinant region, which, however, is not known in advance. Also, this approach does not offer a way to estimate the recombination parameter v. COPY RIGHT © 2011 Institute of Interdisciplinary Business Research JANUARY 2011 VOL 2, NO 9 534
ijcrb.webs.com INTERDISCIPLINARY JOURNAL OF CONTEMPORARY RESEARCH IN BUSINESS 4. Parameter Estimation with Maximum Likelihood: HMM-ML A solution to this problem, proposed by Husmeier and Wright (2001) is a proper maximum likelihood estimation of the parameters so as to maximize with respect to the vector of branch lengths w, the parameters of the nucleotide substitution model θ, and the recombination parameter v. This requires a summation over all state sequences S = (S1,..., SN), that is, over K N terms, and seems to be intractable for all but very short sequence lengths N. However, Husmeier and Wright (2001)showed that by applying the expectation maximization (EM) algorithm (Dempster, Laird, and Rubin 1977, the sparseness of the connectivity in the HMM could be exploited to reduce the computational complexity to the order of K separate tree optimizations. While the application of this scheme outperformed the heuristic approach of McGuire, Wright, and Prentice (2000)it suffers from the shortcoming that the predicted state sequence does not only depend on the data, argmax P(S/D), but also on the parameters, argmax P(S|D,w, θ,v).The fact that these parameters are estimated from the data itself with maximum likelihood renders the approach susceptible to over-fitting. This calls for an independent hypothesis test with parametric bootstrapping, which, however, incurs prohibitively high computational costs, as demonstrated by Larget and Simon (1999) To rephrase this problem, note that hidden Markov models and phylogenetic trees have many similarities with neural networks; in fact, all three models are instances of the more general class of graphical models (Heckermann 1999. Studies on neural networks and graphical models have shown that, for sparse data, maximum likelihood is susceptible to over-fitting, and that the generalization performance is significantly improved with the Bayesian approach. A detailed investigation of this approach can be found in Neal (1996). In a nutshell, maximum likelihood gives only a point estimate of the parameters, which ignores the more detailed information contained in the curvature and (possibly) multimodality of the likelihood landscape. By sampling rather than optimizing parameters, the Bayesian approach captures more information about this landscape, and consequently gives improved and more reliable predictions. A Bayesian approach to phylogenetics without recombination was proposed and tested by Yang and Rannala (1997), Mau, Newton, and Larget (1999) and Larget and Simon (1999). Generalizing this scheme to the presence of recombination requires replacing the single topology-indicating variable by the state sequence S, as discussed in the previous section. The prediction of this state sequence should be based on the posterior probability P(S|D), which requires integrating out the remaining parameters: COPY RIGHT © 2011 Institute of Interdisciplinary Business Research JANUARY 2011 VOL 2, NO 9 535
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