Bulletin of Mathematical Biology

, Volume 75, Issue 12, pp 2529–2550 | Cite as

A Note on Probabilistic Models over Strings: The Linear Algebra Approach

Original Article

Abstract

Probabilistic models over strings have played a key role in developing methods that take into consideration indels as phylogenetically informative events. There is an extensive literature on using automata and transducers on phylogenies to do inference on these probabilistic models, in which an important theoretical question is the complexity of computing the normalization of a class of string-valued graphical models. This question has been investigated using tools from combinatorics, dynamic programming, and graph theory, and has practical applications in Bayesian phylogenetics. In this work, we revisit this theoretical question from a different point of view, based on linear algebra. The main contribution is a set of results based on this linear algebra view that facilitate the analysis and design of inference algorithms on string-valued graphical models. As an illustration, we use this method to give a new elementary proof of a known result on the complexity of inference on the “TKF91” model, a well-known probabilistic model over strings. Compared to previous work, our proving method is easier to extend to other models, since it relies on a novel weak condition, triangular transducers, which is easy to establish in practice. The linear algebra view provides a concise way of describing transducer algorithms and their compositions, opens the possibility of transferring fast linear algebra libraries (for example, based on GPUs), as well as low rank matrix approximation methods, to string-valued inference problems.

Keywords

Indel Alignment Probabilistic models TKF91 String transducers Automata Graphical models Phylogenetics Factor graphs 

References

  1. Airoldi, E. M. (2007). Getting started in probabilistic graphical models. PLoS Comput. Biol., 3(12). Google Scholar
  2. Bishop, C. M. (2006). Pattern recognition and machine learning (pp. 359–422). Berlin: Springer. Chap. 8. MATHGoogle Scholar
  3. Bouchard-Côté, A., & Jordan, M. I. (2012). Evolutionary inference via the Poisson indel process. Proc. Nat. Acad. Sci. USA. doi:10.1073/pnas.1220450110. Google Scholar
  4. Bouchard-Côté, A., Jordan, M. I., & Klein, D. (2009). Efficient inference in phylogenetic InDel trees. In Advances in neural information processing systems (Vol. 21). Google Scholar
  5. Bouchard-Côté, A., Sankararaman, S., & Jordan, M. I. (2012). Phylogenetic inference via sequential Monte Carlo. Syst. Biol., 61, 579–593. CrossRefGoogle Scholar
  6. Bradley, R. K., & Holmes, I. (2007). Transducers: an emerging probabilistic framework for modeling indels on trees. Bioinformatics, 23(23), 3258–3262. CrossRefGoogle Scholar
  7. Daskalakis, C., & Roch, S. (2012). Alignment-free phylogenetic reconstruction: Sample complexity via a branching process analysis. Ann. Appl. Probab. Google Scholar
  8. Dreyer, M., Smith, J. R., & Eisner, J. (2008). Latent-variable modeling of string transductions with finite-state methods. In Proceedings of EMNLP 2008. Google Scholar
  9. Droste, M., & Kuich, W. (2009). Handbook of weighted automata. Monographs in theoretical computer science. Berlin: Springer. Chap. 1. CrossRefMATHGoogle Scholar
  10. Eilenberg, S. (1974). Automata, languages and machines (Vol. A). San Diego: Academic Press. MATHGoogle Scholar
  11. Felsenstein, J. (1981). Evolutionary trees from DNA sequences: a maximum likelihood approach. J. Mol. Evol., 17, 368–376. CrossRefGoogle Scholar
  12. Felsenstein, J. (2003). Inferring phylogenies. Sunderland: Sinauer Associates. Google Scholar
  13. Fernandez, P., Plateau, B., & Stewart, W. J. (1998). Optimizing tensor product computations in stochastic automata networks. RAIRO. Rech. Opér., 32(3), 325–351. Google Scholar
  14. Görür, D., & Teh, Y. W. (2008). An efficient sequential Monte-Carlo algorithm for coalescent clustering. In Advances in neural information processing (pp. 521–528). Red Hook: Curran Associates. Google Scholar
  15. Hein, J. (1990). A unified approach to phylogenies and alignments. Methods Enzymol., 183, 625–944. Google Scholar
  16. Hein, J. (2000). A generalisation of the Thorne–Kishino–Felsenstein model of statistical alignment to k sequences related by a binary tree. In Pac. symp. biocomput. (pp. 179–190). Google Scholar
  17. Hein, J. (2001). An algorithm for statistical alignment of sequences related by a binary tree. In Pac. symp. biocomput. (pp. 179–190). Google Scholar
  18. Hein, J., Jensen, J., & Pedersen, C. (2003). Recursions for statistical multiple alignment. Proc. Natl. Acad. Sci. USA, 100(25), 14960–14965. CrossRefGoogle Scholar
  19. Higdon, D. M. (1998). Auxiliary variable methods for Markov Chain Monte Carlo with applications. J. Am. Stat. Assoc., 93(442), 585–595. CrossRefMATHGoogle Scholar
  20. Holmes, I. (2003). Using guide trees to construct multiple-sequence evolutionary hmms. Bioinformatics, 19(1), 147–157. CrossRefGoogle Scholar
  21. Holmes, I. (2007). Phylocomposer and phylodirector: analysis and visualization of transducer indel models. Bioinformatics, 23(23), 3263–3264. CrossRefGoogle Scholar
  22. Holmes, I., & Bruno, W. J. (2001). Evolutionary HMM: a Bayesian approach to multiple alignment. Bioinformatics, 17, 803–820. CrossRefGoogle Scholar
  23. Holmes, I., & Rubin, G. M. (2002). An expectation maximization algorithm for training hidden substitution models. J. Mol. Biol. Google Scholar
  24. Jensen, J., & Hein, J. (2002). Gibbs sampler for statistical multiple alignment (Technical report). Dept of Theor Stat, University of Aarhus. Google Scholar
  25. Jordan, M. I. (2004). Graphical models. Stat. Sci., 19, 140–155. CrossRefMATHGoogle Scholar
  26. Kawakita, A., Sota, T., Ascher, J. S., Ito, M., Tanaka, H., & Kato, M. (2003). Evolution and phylogenetic utility of alignment gaps within intron sequences of three nuclear genes in bumble bees (Bombus). Mol. Biol. Evol., 20(1), 87–92. CrossRefGoogle Scholar
  27. Knudsen, B., & Miyamoto, M. (2003). Sequence alignments and pair hidden Markov models using evolutionary history. J. Mol. Biol., 333, 453–460. CrossRefGoogle Scholar
  28. Langville, A. N., & Stewart, W. J. (2004). The Kronecker product and stochastic automata networks. J. Comput. Appl. Math., 167(2), 429–447. MathSciNetCrossRefMATHGoogle Scholar
  29. Lunter, G., Miklós, I., Drummond, A., Jensen, J., & Hein, J. (2005). Bayesian coestimation of phylogeny and sequence alignment. BMC Bioinform., 6(1), 83. CrossRefGoogle Scholar
  30. Metzler, D., Fleissner, R., Wakolbinger, A., & von Haeseler, A. (2001). Assessing variability by joint sampling of alignments and mutation rates. J. Mol. Biol.. Google Scholar
  31. Miklós, I., & Toroczkai, Z. (2001). An improved model for statistical alignment. In First workshop on algorithms in bioinformatics, Berlin: Springer. Google Scholar
  32. Miklós, I., Drummond, A., Lunter, G., & Hein, J. (2003a). Bayesian phylogenetic inference under a statistical insertion–deletion model. In Algorithms in bioinformatics, Berlin: Springer. Google Scholar
  33. Miklós, I., Song, Y. S., Lunter, G. A., & Hein, J. (2003b). An efficient algorithm for statistical multiple alignment on arbitrary phylogenetic trees. J. Comput. Biol., 10, 869–889. CrossRefGoogle Scholar
  34. Miklós, I., Lunter, G. A., & Holmes, I. (2004). A long indel model for evolutionary sequence alignment. Mol. Biol. Evol., 21(3), 529–540. CrossRefGoogle Scholar
  35. Mingming, S. (2012). Gpumatrix library. Google Scholar
  36. Mohri, M. (2002). Generic epsilon-removal and input epsilon-normalization algorithms for weighted transducers. Int. J. Found. Comput. Sci., 13(1), 129–143. MathSciNetCrossRefMATHGoogle Scholar
  37. Mohri, M. (2009). Handbook of weighted automata. Monographs in theoretical computer science. Berlin: Springer. Chap. 6. Google Scholar
  38. Novák, Á., Miklós, I., Lyngsoe, R., & Hein, J. (2008). Statalign: an extendable software package for joint Bayesian estimation of alignments and evolutionary trees. Bioinformatics, 24, 2403–2404. CrossRefGoogle Scholar
  39. Redelings, B. D., & Suchard, M. A. (2005). Joint Bayesian estimation of alignment and phylogeny. Syst. Biol., 54(3), 401–418. CrossRefGoogle Scholar
  40. Redelings, B. D., & Suchard, M. A. (2007). Incorporating indel information into phylogeny estimation for rapidly emerging pathogens. BMC Evol. Biol., 7(40). Google Scholar
  41. Rivas, E. (2005). Evolutionary models for insertions and deletions in a probabilistic modeling framework. BMC Bioinform., 6(1), 63. CrossRefGoogle Scholar
  42. Satija, R., Pachter, L., & Hein, J. (2008). Combining statistical alignment and phylogenetic footprinting to detect regulatory elements. Bioinformatics, 24, 1236–1242. CrossRefGoogle Scholar
  43. Schützenberger, M. P. (1961). On the definition of a family of automata. Inf. Control, 4, 245–270. CrossRefMATHGoogle Scholar
  44. Song, Y. S. (2006). A sufficient condition for reducing recursions in hidden Markov models. Bull. Math. Biol., 68, 361–384. MathSciNetCrossRefGoogle Scholar
  45. Steel, M., & Hein, J. (2001). Applying the Thorne–Kishino–Felsenstein model to sequence evolution on a star-shaped tree. Appl. Math. Lett., 14, 679–684. MathSciNetCrossRefMATHGoogle Scholar
  46. Teh, Y. W., Daume, H. III, & Roy, D. M. (2008). Bayesian agglomerative clustering with coalescents. In Advances in neural information processing (pp. 1473–1480). Cambridge: MIT Press. Google Scholar
  47. Thorne, J. L., Kishino, H., & Felsenstein, J. (1991). An evolutionary model for maximum likelihood alignment of DNA sequences. J. Mol. Evol., 33, 114–124. CrossRefGoogle Scholar
  48. Thorne, J. L., Kishino, H., & Felsenstein, J. (1992). Inching toward reality: an improved likelihood model of sequence evolution. J. Mol. Evol., 34, 3–16. CrossRefGoogle Scholar
  49. Westesson, O., Lunter, G., Paten, B., & Holmes, I. (2011). Phylogenetic automata, pruning, and multiple alignment. Preprint, arXiv:1103.4347.
  50. Westesson, O., Lunter, G., Paten, B., & Holmes, I. (2012). Accurate reconstruction of insertion-deletion histories by statistical phylogenetics. PLoS ONE, 7(4), e34572. CrossRefGoogle Scholar
  51. Whaley, R.C., Petitet, A., & Dongarra, J. J. (2001). Automated empirical optimization of software and the ATLAS project. Parallel Comput., 27(1–2), 3–35. CrossRefMATHGoogle Scholar
  52. Williams, V. V. (2012). Multiplying matrices faster than Coppersmith–Winograd. In STOC. Google Scholar
  53. Wong, K. M., Suchard, M. A., & Huelsenbeck, J. P. (2008). Alignment uncertainty and genomic analysis. Science, 319(5862), 473–476. MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  1. 1.Department of StatisticsThe University of British ColumbiaVancouverCanada

Personalised recommendations