Embedding of Markov matrices for $d\leqslant 4$
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910510470922240 |
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| author | Baake, Michael Sumner, Jeremy |
| author_facet | Baake, Michael Sumner, Jeremy |
| contents | The embedding problem of Markov matrices in Markov semigroups is a classic problem that regained a lot of impetus and activities through recent needs in phylogeny and population genetics. Here, we give an account for dimensions $d\leqslant 4$, including a complete and simplified treatment of the case $d=3$, and derive the results in a systematic fashion, with an eye on the potential applications.
Further, we reconsider the setup of the corresponding problem for time-inhomogeneous Markov chains, which is needed for real-world applications because transition rates need not be constant over time. Additional cases of this more general embedding occur for any $d\geqslant 3$. We review the known case of $d=3$ and describe the setting for future work on $d=4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_02596 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Embedding of Markov matrices for $d\leqslant 4$ Baake, Michael Sumner, Jeremy Probability Populations and Evolution 60J10, 60J27, 92D10 The embedding problem of Markov matrices in Markov semigroups is a classic problem that regained a lot of impetus and activities through recent needs in phylogeny and population genetics. Here, we give an account for dimensions $d\leqslant 4$, including a complete and simplified treatment of the case $d=3$, and derive the results in a systematic fashion, with an eye on the potential applications. Further, we reconsider the setup of the corresponding problem for time-inhomogeneous Markov chains, which is needed for real-world applications because transition rates need not be constant over time. Additional cases of this more general embedding occur for any $d\geqslant 3$. We review the known case of $d=3$ and describe the setting for future work on $d=4$. |
| title | Embedding of Markov matrices for $d\leqslant 4$ |
| topic | Probability Populations and Evolution 60J10, 60J27, 92D10 |
| url | https://arxiv.org/abs/2311.02596 |