{
  "abstract": "Computing phylogenetic likelihoods becomes challenging when the continuous-time Markov chain has a large state space. At each branch, pruning algorithms require terms of the form $e^{Q t}\\cdot B$, where $Q$ is an $S \\times S$ rate matrix (e.g., for $S$ states), $t$ is a branch length, and the columns of matrix $B$ (e.g., $B \\in \\mathbb{R}_{+}^{S \\times L}$) are partial-likelihood vectors for $L$ site patterns. Many implementations calculate the dense transition matrix $P(t)=e^{Q t}$ via eigendecomposition and multiply it by $B$. This strategy is efficient for small state spaces, but it can be inefficient for large sparse rate matrices because forming $P(t)$ or its eigen vector matrices destroys sparsity and its cost may not be well amortized when $L$ is small. We implement an alternative likelihood kernel based on the Al-Mohy and Higham (2011) algorithm for computing the action of the matrix exponential directly, avoiding explicit construction of $P(t)$ or its eigen vector matrices. The resulting method takes full advantage of the sparsity of $Q$. We implement this method as a CPU-based extension to the BEAGLE library and evaluate it on two large-state molecular models: a Markov-modulated codon model of heterotachy with $M$ selective regimes and $61\\cdot M$ possible states, and a codon-pair model of interlocus gene conversion between two paralogs with $61^2=3\\,721$ possible states. Our results characterize when action-based likelihood computation improves performance over calculating $e^{Q t}$ directly. We show that this method enables fast, scalable inference for complex models, removing a key computational barrier to the study of evolution with large state space models.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03873",
  "author_total": 12,
  "authors": [
    {
      "name": "Benjamin D Redelings"
    },
    {
      "name": "April Wright"
    },
    {
      "name": "Karthik Gangavarapu"
    },
    {
      "name": "Yucai Shao"
    },
    {
      "name": "Filippo Monti"
    },
    {
      "name": "Wu-Min Deng"
    },
    {
      "name": "Shuo Su"
    },
    {
      "name": "Andrew Rambaut"
    },
    {
      "name": "Guy Baele"
    },
    {
      "name": "Philippe Lemey"
    },
    {
      "name": "Marc A. Suchard"
    },
    {
      "name": "Xiang Ji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1f3t5dscxak29.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03873",
  "fulltext_url": null,
  "id": "1f3t5dscxak29",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03873",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "q-bio.PE"
    ],
    [
      "arxiv",
      "stat.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03873",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1f3t5dscxak29.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:30Z",
      "native_id": "2611.03873",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03873"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.PE",
      "subfield": "evolutionary-biology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.CO",
      "subfield": "statistics"
    }
  ],
  "title": "BEAGLE in Action: Phylogenetic Likelihoods for Very Large State Spaces",
  "title_text": "BEAGLE in Action: Phylogenetic Likelihoods for Very Large State Spaces",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:10Z",
  "url": "https://research.draconitic.com/paper/1f3t5dscxak29",
  "venue": "arXiv",
  "versions": []
}