Notes on the maximum likelihood estimation of haplotype frequencies

Shuhei Mano*, N. Yasuda, T. Katoh, K. Tounai, H. Inoko, T. Imanishi, G. Tamiya, T. Gojobori

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


The maximum likelihood estimation (MLE) is one of the most popular ways to estimate haplotype frequencies of a population with genotype data whose linkage phases are unknown. The MLE is commonly implemented in the use of the Expectation-Maximization (EM) algorithm. It is known that the EM algorithm carries the risk that an estimator may converge erroneously to one of the local maxima or saddle points of the likelihood surface, resulting in serious errors in the MLE of haplotype frequencies. In this note, by theoretical treatments we present the necessary and sufficient conditions that the local maxima or saddle points on the likelihood surface appear. As a rule of thumb, that the difference between the coupling and repulsive haplotype frequencies in phase known individuals is 3/2 times larger than the frequency of phase ambiguous individuals is the sufficient condition that the likelihood surface is unimodal. Moreover, we present the analytic solution to the biallelic two-locus problem, and construct a general algorithm to obtain the global maximum.

Original languageEnglish (US)
Pages (from-to)257-264
Number of pages8
JournalAnnals of Human Genetics
Issue number3
StatePublished - May 2004
Externally publishedYes


  • Analytic solution
  • EM algorithm
  • Haplotype frequency estimation
  • Likelihood surface
  • Maximum likelihood estimation

ASJC Scopus subject areas

  • Genetics
  • Genetics(clinical)


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