In 1922 R. A. Fisher introduced the method of maximum likelihood. He first presented the numerical procedure in 1912. This paper considers Fisher’s changing justifications for the method, the concepts he developed around it \vZincluding likelihood, sufficiency, effi- ciency and information. and the approaches he discarded \vZincluding inverse probability..

The application of maximum likelihood techniques to the estimation of evolutionary trees from nucleic acid sequence data is discussed. A computationally feasible method for finding such maximum likelihood estimates is developed, and a computer program is available. This method has advantages over the traditional parsimony algorithms, which can give misleading results if rates of evolution differ in different lineages. It also allows the testing of hypotheses about the constancy of evolutionary rates by likelihood ratio tests, and gives rough indication of the error of ;the estimate of the tree.

Using a maximum-likelihood formalism, we have developed a method with which to reconstruct the sequences of ancestral proteins. Our approach allows the calculation of not only the most probable ancestral sequence but also of the probability of any amino acid at any given node in the evolutionary tree. Because we consider evolution on the amino acid level, we are better able to include effects of evolutionary pressure and take advantage of structural information about the protein through the use of mutation matrices that depend on secondary structure and surface accessibility. The computational complexity of this method scales linearly with the number of homologous proteins used to reconstruct the ancestral sequence.

A statistical method was developed for reconstructing the nucleotide or amino acid sequences of extinct ancestors, given the phylogeny and sequences of the extant species. A model of nucleotide or amino acid substitution was employed to analyze data of the present-day sequences, and maximum likelihood estimates of parameters such as branch lengths were used to compare the posterior probabilities of assignments of character states (nucleotides or amino acids) to interior nodes of the tree; the assignment having the highest probability was the best reconstruction at the site. The lysozyme c sequences of six mammals were analyzed by using the likelihood and parsimony methods. The new likelihood-based method was found to be superior to the parsimony method. The probability that the amino acids for all interior nodes at a site reconstructed by the new method are correct was calculated to be 0.91, 0.86, and 0.73 for all, variable, and parsimony-informative sites, respectively, whereas the corresponding probabilities for the parsimony method were 0.84, 0.76, and 0.51, respectively. The probability that an amino acid in an ancestral sequence is correctly reconstructed by the likelihood analysis ranged from 91.3 to 98.7% for the four ancestral sequences.

One of the strengths of the maximum likelihood method of phylogenetic estimation is the ease with which hypotheses can be formulated and tested. Maximum likelihood analysis of DNA and amino acid sequence data has been made practical with recent advances in models of DNA substitution, computer programs, and computational speed. Here, we describe the maximum likelihood method and the recent improvements in models of substitution. We also describe how likelihood ratio tests of a variety of biological hypotheses can be formulated and tested using computer simulation to generate the null distribution of the likelihood ratio test statistic.

The increase in the number of large data sets and the complexity of current probabilistic sequence evolution models necessitates fast and reliable phylogeny reconstruction methods. We describe a new approach, based on the maximum- likelihood principle, which clearly satisfies these requirements. The core of this method is a simple hill-climbing algorithm that adjusts tree topology and branch lengths simultaneously. This algorithm starts from an initial tree built by a fast distance-based method and modifies this tree to improve its likelihood at each iteration. Due to this simultaneous adjustment of the topology and branch lengths, only a few iterations are sufficient to reach an optimum. We used extensive and realistic computer simulations to show that the topological accuracy of this new method is at least as high as that of the existing maximum-likelihood programs and much higher than the performance of distance-based and parsimony approaches. The reduction of computing time is dramatic in comparison with other maximum-likelihood packages, while the likelihood maximization ability tends to be higher. For example, only 12 min were required on a standard personal computer to analyze a data set consisting of 500 rbcL sequences with 1,428 base pairs from plant plastids, thus reaching a speed of the same order as some popular distance-based and parsimony algorithms. This new method is implemented in the PHYML program, which is freely available on our web page: http://www.lirmm.fr/w3ifa/MAAS/.

Phylogenetic inference from amino acid sequence data uses mainly empirical models of amino acid replacement and is therefore dependent on those models. Two of the more widely used models, the Dayhoff and JTT models, are estimated using similar methods that can utilize large numbers of sequences from many unrelated protein families but are somewhat unsatisfactory because they rely on assumptions that may lead to systematic error and discard a large amount of the information within the sequences. The alternative method of maximum-likelihood estimation may utilize the information in the sequence data more efficiently and suffers from no systematic error, but it has previously been applicable to relatively few sequences related by a single phylogenetic tree. Here, we combine the best attributes of these two methods using an approximate maximum-likelihood method. We implemented this approach to estimate a new model of amino acid replacement from a database of globular protein sequences comprising 3,905 amino acid sequences split into 182 protein families. While the new model has an overall structure similar to those of other commonly used models, there are significant differences. The new model outperforms the Dayhoff and JTT models with respect to maximum-likelihood values for a large majority of the protein families in our database. This suggests that it provides a better overall fit to the evolutionary process in globular proteins and may lead to more accurate phylogenetic tree estimates. Potentially, this matrix, and the methods used to generate it, may also be useful in other areas of research, such as biological sequence database searching, sequence alignment, and protein structure prediction, for which an accurate description of amino acid replacement is required.

Felsenstein?s maximum-likelihood approach for inferring phylogeny from DNA sequences assumes that the rate of nucleotide substitution is constant over different nucleotide sites. This assumption is sometimes unrealistic, as has been revealed by analysis of real sequence data. In the present paper Felsenstein?s method is extended to the case where substitution rates over sites are described by the gamma distribution. A numerical example is presented to show that the method fits the data better than do previous models.