Maximum Likelihood

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model by finding the parameter values that maximize the likelihood function — the probability of observing the given data under the model. In phylogenetics, maximum likelihood is the dominant framework for reconstructing evolutionary trees: it searches the space of possible phylogenetic trees and finds the tree that makes the observed sequence data most probable under a specified model of molecular evolution.

The strength of maximum likelihood is its statistical consistency and its explicit modeling framework: it forces researchers to state their assumptions about substitution rates, base frequencies, and rate heterogeneity. The weakness is that the method is only as good as its model, and the models are always wrong. Biological reality — recombination, horizontal transfer, selection, codon bias — routinely violates the assumptions of standard evolutionary models. A maximum likelihood tree with high support may be the best tree under a biologically unrealistic model, which is a form of precision without accuracy.

Maximum likelihood has been largely supplanted by Bayesian inference in some domains not because Bayesian methods are better, but because they provide a more natural framework for incorporating prior knowledge and comparing models through model selection criteria. The debate between frequentist maximum likelihood and Bayesian phylogenetics is ultimately a debate about whether the best tree is the one that maximizes probability of data or the one that maximizes posterior probability given data and prior assumptions. Both are valid formalisms; neither escapes the problem that the model is a fiction.

Maximum likelihood phylogenetics represents the triumph of statistical formalism over biological realism. The method is mathematically impeccable and biologically naive — a perfect example of what happens when elegance in the model is mistaken for truth in the world.