{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:25:52Z","timestamp":1760243152358,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2015,12,11]],"date-time":"2015-12-11T00:00:00Z","timestamp":1449792000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>A key parameter in population genetics is the scaled mutation rate                                        \u03b8               =               4               N               \u03bc                                 , where N is the effective haploid population size and \u03bc is the mutation rate per haplotype per generation. While exact likelihood inference is notoriously difficult in population genetics, we propose a novel approach to compute a first order accurate likelihood of \u03b8 that is based on dynamic programming under the infinite sites model without recombination. The parameter \u03b8 may be either constant, i.e., time-independent, or time-dependent, which allows for changes of demography and deviations from neutral equilibrium. For time-independent \u03b8, the performance is compared to the approach in Griffiths and Tavar\u00e9\u2019s work \u201cSimulating Probability Distributions in the Coalescent\u201d (Theor. Popul. Biol. 1994, 46, 131\u2013159) that is based on importance sampling and implemented in the \u201cgenetree\u201d program. Roughly, the proposed method is computationally fast when                                        n               \u00d7               \u03b8               &lt;               100                                 , where n is the sample size. For time-dependent                                        \u03b8               (               t               )                                 , we analyze a simple demographic model with a single change in                                        \u03b8               (               t               )                                 . In this case, the ancestral and current \u03b8 need to be estimated, as well as the time of change. To our knowledge, this is the first accurate computation of a likelihood in the infinite sites model with non-equilibrium demography.<\/jats:p>","DOI":"10.3390\/computation3040701","type":"journal-article","created":{"date-parts":[[2015,12,14]],"date-time":"2015-12-14T02:57:29Z","timestamp":1450061849000},"page":"701-713","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Exact Likelihood Calculation under the Infinite Sites Model"],"prefix":"10.3390","volume":"3","author":[{"given":"Muhammad","family":"Faisal","sequence":"first","affiliation":[{"name":"Faculty of Health Studies, University of Bradford, Bradford BD71DP, UK"},{"name":"Bradford Institute for Health Research, Bradford Teaching Hospitals NHS Foundation Trust, Bradford BD96RJ, UK"},{"name":"Department of Statistics and Operations Research, University of Vienna, 1090 Vienna, Austria"}]},{"given":"Andreas","family":"Futschik","sequence":"additional","affiliation":[{"name":"Institute of Applied Statistics, Johannes Kepler University Linz, 4040 Linz, Austria"}]},{"given":"Claus","family":"Vogl","sequence":"additional","affiliation":[{"name":"Institute of Animal Breeding and Genetics, University of Veterinary Medicine Vienna, Veterin\u00e4rplatz 1, 1210 Vienna, Austria"}]}],"member":"1968","published-online":{"date-parts":[[2015,12,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1093\/genetics\/61.4.893","article-title":"The number of heterozygous nucleotide sites maintained in a finite population due to steady flux of mutations","volume":"61","author":"Kimura","year":"1969","journal-title":"Genetics"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hein, J., Schierup, M.H., and Wiuf, C. (2005). Gene Genealogies, Variation and Evolution: A Primer in Coalescent Theory, Oxford University Press.","DOI":"10.1093\/oso\/9780198529958.001.0001"},{"key":"ref_3","unstructured":"Wakeley, J. (2009). Coalescent Theory: An Introduction, Roberts & Co. Publishers."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1969","DOI":"10.1093\/molbev\/mss075","article-title":"Bayesian Phylogenetics with BEAUti and the BEAST 1.7","volume":"29","author":"Drummond","year":"2012","journal-title":"Mol. Biol. Evol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1093\/bioinformatics\/btk051","article-title":"LAMARC 2.0: Maximum likelihood and Bayesian estimation of population parameters","volume":"22","author":"Kuhner","year":"2006","journal-title":"Bioinformatics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1214\/ss\/1177010378","article-title":"Ancestral Inference in Population Genetics","volume":"9","author":"Griffiths","year":"1994","journal-title":"Stat. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1006\/tpbi.1994.1023","article-title":"Simulating Probability Distributions in the Coalescent","volume":"46","author":"Griffiths","year":"1994","journal-title":"Theor. Popul. Biol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1109\/TCBB.2010.2","article-title":"Exact computation of coalescent likelihood for panmictic and subdivided populations under the infinite sites model","volume":"7","author":"Wu","year":"2010","journal-title":"IEEE\/ACM Trans. Comput. Biol. Bioinform."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1111\/1467-9868.00254","article-title":"Inference in molecular population genetics","volume":"62","author":"Stephens","year":"2000","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1093\/bioinformatics\/18.2.337","article-title":"Generating samples under a Wright-Fisher neutral model of genetic variation","volume":"18","author":"Hudson","year":"2002","journal-title":"Bioinformatics"},{"key":"ref_11","unstructured":"Press, W., Teukolsky, S., Vetterling, W., and Flannery, B. (2007). Numerical Recipes: The Art of Scientific Computing, Cambridge University Press. [3rd ed.]. Chapter 10.4."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1214\/lnms\/1215455552","article-title":"Likelihoods on Coalescents: A Monte Carlo Sampling Approach to Inferring Parameters from Population Samples of Molecular Data","volume":"33","author":"Felsenstein","year":"1999","journal-title":"Lect. Notes Monogr. Ser."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/3\/4\/701\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:53:54Z","timestamp":1760216034000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/3\/4\/701"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,12,11]]},"references-count":12,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2015,12]]}},"alternative-id":["computation3040701"],"URL":"https:\/\/doi.org\/10.3390\/computation3040701","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2015,12,11]]}}}