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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