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With the advent of high-throughput sequencing, recent Bayesian phylogenetic studies have analyzed hundreds to thousands of sequences. Such large-scale analyses challenge divergence time reconstruction by requiring inference on highly correlated internal node heights that often become computationally infeasible. To overcome this limitation, we explore a ratio transformation that maps the original $N-1$ internal node heights into a space of one height parameter and $N-2$ ratio parameters. To make the analyses scalable, we develop a collection of linear-time algorithms to compute the gradient and Jacobian-associated terms of the log-likelihood with respect to these ratios. We then apply Hamiltonian Monte Carlo sampling with the ratio transform in a Bayesian framework to learn the divergence times in 4 pathogenic viruses (West Nile virus, rabies virus, Lassa virus, and Ebola virus) and the coralline red algae. Our method both resolves a mixing issue in the West Nile virus example and improves inference efficiency by at least 5-fold for the Lassa and rabies virus examples as well as for the algae example. Our method now also makes it computationally feasible to incorporate mixed-effects molecular clock models for the Ebola virus example, confirms the findings from the original study, and reveals clearer multimodal distributions of the divergence times of some clades of interest.<\/jats:p>","DOI":"10.1093\/sysbio\/syad039","type":"journal-article","created":{"date-parts":[[2023,7,17]],"date-time":"2023-07-17T15:12:17Z","timestamp":1689606737000},"page":"1136-1153","source":"Crossref","is-referenced-by-count":9,"title":["Scalable Bayesian Divergence Time Estimation With Ratio Transformations"],"prefix":"10.1093","volume":"72","author":[{"given":"Xiang","family":"Ji","sequence":"first","affiliation":[{"name":"Department of Mathematics, School of Science & Engineering, Tulane University , 6823 St. Charles Avenue, New Orleans, LA 70118 , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8381-6587","authenticated-orcid":false,"given":"Alexander A","family":"Fisher","sequence":"additional","affiliation":[{"name":"Department of Statistical Science , Duke University, 214 Old Chemistry, Durham, NC 27708 , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shuo","family":"Su","sequence":"additional","affiliation":[{"name":"MOE International Joint Collaborative Research Laboratory for Animal Health & Food Safety, Jiangsu Engineering Laboratory of Animal Immunology, Institute of Immunology, College of Veterinary Medicine, Nanjing Agricultural University , No. 1 Weigang, Xiaolingwei District, Nanjing, Jiangsu 210095 , China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jeffrey L","family":"Thorne","sequence":"additional","affiliation":[{"name":"Bioinformatics Research Center, North Carolina State University , Raleigh, NC , USA"},{"name":"Department of Statistics, North Carolina State University , Raleigh, NC , USA"},{"name":"Department of Biological Sciences, North Carolina State University , Ricks Hall, 1 Lampe Dr, Raleigh, NC 27607 , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9476-149X","authenticated-orcid":false,"given":"Barney","family":"Potter","sequence":"additional","affiliation":[{"name":"Department of Microbiology, Immunology and Transplantation, Rega Institute , Herestraat 49, 3000 Leuven , Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Philippe","family":"Lemey","sequence":"additional","affiliation":[{"name":"Department of Microbiology, Immunology and Transplantation, Rega Institute , Herestraat 49, 3000 Leuven , Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1915-7732","authenticated-orcid":false,"given":"Guy","family":"Baele","sequence":"additional","affiliation":[{"name":"Department of Microbiology, Immunology and Transplantation, Rega Institute , Herestraat 49, 3000 Leuven , Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9818-479X","authenticated-orcid":false,"given":"Marc A","family":"Suchard","sequence":"additional","affiliation":[{"name":"Department of Biomathematics, David Geffen School of Medicine, University of California Los Angeles , Los Angeles, CA , USA"},{"name":"Department of Human Genetics, David Geffen School of Medicine, University of California Los Angeles , Los Angeles, CA , USA"},{"name":"Department of Biostatistics, Fielding School of Public Health, University of California Los Angeles , 695 Charles E Young Dr S, Los Angeles, CA 90095 , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2023,7,17]]},"reference":[{"issue":"1","key":"2023111006562002400_CIT0001","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s12862-015-0408-z","article-title":"Time dependency of foamy virus evolutionary rate estimates","volume":"15","author":"Aiewsakun","year":"2015","journal-title":"BMC Evolut. 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