Algebraic Statistics

Sullivant, Seth

doi:10.1090/gsm/194

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Algebraic Statistics

About this Title

Seth Sullivant, North Carolina State University, Raleigh, NC

Publication: Graduate Studies in Mathematics
Publication Year: 2018; Volume 194
ISBNs: 978-1-4704-3517-2 (print); 978-1-4704-4980-3 (online)
DOI: https://doi.org/10.1090/gsm/194
MathSciNet review: MR3838364
MSC: Primary 62-01; Secondary 13P25, 14P10, 60J10, 62F03, 62H17, 92D15

PDF View full volume as PDF

Read more about this volume

Carlos Améndola, Nathan Bliss, Isaac Burke, Courtney R. Gibbons, Martin Helmer, Serkan Hoşten, Evan D. Nash, Jose Israel Rodriguez, and Daniel Smolkin, The maximum likelihood degree of toric varieties, arXiv:1703.02251, 2017.
Carlos Améndola, Jean-Charles Faugère, and Bernd Sturmfels, Moment varieties of Gaussian mixtures, J. Algebr. Stat. 7 (2016), no. 1, 14–28. MR 3529332, DOI 10.18409/jas.v7i1.42
Alan Agresti, Categorical data analysis, 3rd ed., Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2013. MR 3087436
V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from the Russian by Hugh Porteous; Translation revised by the authors and James Montaldi. MR 966191, DOI 10.1007/978-1-4612-3940-6
David F. Andrews and A. M. Herzberg, Data, Springer, New York, 1985.
Satoshi Aoki, Hisayuki Hara, and Akimichi Takemura, Markov bases in algebraic statistics, Springer Series in Statistics, Springer, New York, 2012. MR 2961912, DOI 10.1007/978-1-4614-3719-2
Federico Ardila and Caroline J. Klivans, The Bergman complex of a matroid and phylogenetic trees, J. Combin. Theory Ser. B 96 (2006), no. 1, 38–49. MR 2185977, DOI 10.1016/j.jctb.2005.06.004
Hirotugu Akaike, A new look at the statistical model identification, IEEE Trans. Automatic Control AC-19 (1974), 716–723. System identification and time-series analysis. MR 0423716, DOI 10.1109/tac.1974.1100705
Srinivas M. Aji and Robert J. McEliece, The generalized distributive law, IEEE Trans. Inform. Theory 46 (2000), no. 2, 325–343. MR 1748973, DOI 10.1109/18.825794
Steen A. Andersson, David Madigan, and Michael D. Perlman, A characterization of Markov equivalence classes for acyclic digraphs, Ann. Statist. 25 (1997), no. 2, 505–541. MR 1439312, DOI 10.1214/aos/1031833662
Steen A. Andersson, David Madigan, and Michael D. Perlman, Alternative Markov properties for chain graphs, Scand. J. Statist. 28 (2001), no. 1, 33–85. MR 1844349, DOI 10.1111/1467-9469.00224
Elizabeth S. Allman, Catherine Matias, and John A. Rhodes, Identifiability of parameters in latent structure models with many observed variables, Ann. Statist. 37 (2009), no. 6A, 3099–3132. MR 2549554, DOI 10.1214/09-AOS689
George E. Andrews, A lower bound for the volume of strictly convex bodies with many boundary lattice points, Trans. Amer. Math. Soc. 106 (1963), 270–279. MR 143105, DOI 10.1090/S0002-9947-1963-0143105-7
Elizabeth Allman, Sonja Petrovic, John Rhodes, and Seth Sullivant, Identifiability of two-tree mixtures for group-based models, IEEE/ACM Transactions on Computational Biology and Bioinformatics 8 (2011), 710–722.
Elizabeth S. Allman and John A. Rhodes, The identifiability of tree topology for phylogenetic models, including covarion and mixture models, J. Comput. Biol. 13 (2006), no. 5, 1101–1113. MR 2255411, DOI 10.1089/cmb.2006.13.1101
Elizabeth S. Allman and John A. Rhodes, Phylogenetic ideals and varieties for the general Markov model, Adv. in Appl. Math. 40 (2008), no. 2, 127–148. MR 2388607, DOI 10.1016/j.aam.2006.10.002
Federico Ardila, Subdominant matroid ultrametrics, Ann. Comb. 8 (2004), no. 4, 379–389. MR 2112691, DOI 10.1007/s00026-004-0227-1
Elizabeth S. Allman, John A. Rhodes, and Seth Sullivant, Statistically consistent $k$-mer methods for phylogenetic tree reconstruction, J. Comput. Biol. 24 (2017), no. 2, 153–171. MR 3607847, DOI 10.1089/cmb.2015.0216
Elizabeth S. Allman, John A. Rhodes, Elena Stanghellini, and Marco Valtorta, Parameter identifiability of discrete Bayesian networks with hidden variables, J. Causal Inference 3 (2015), 189–205.
Elizabeth S. Allman, John A. Rhodes, Bernd Sturmfels, and Piotr Zwiernik, Tensors of nonnegative rank two, Linear Algebra Appl. 473 (2015), 37–53. MR 3338324, DOI 10.1016/j.laa.2013.10.046
Elizabeth S. Allman, John A. Rhodes, and Amelia Taylor, A semialgebraic description of the general Markov model on phylogenetic trees, SIAM J. Discrete Math. 28 (2014), no. 2, 736–755. MR 3206983, DOI 10.1137/120901568
Satoshi Aoki and Akimichi Takemura, Minimal basis for a connected Markov chain over $3\times 3\times K$ contingency tables with fixed two-dimensional marginals, Aust. N. Z. J. Stat. 45 (2003), no. 2, 229–249. MR 1983834, DOI 10.1111/1467-842X.00278
Miki Aoyagi and Sumio Watanabe, Stochastic complexities of reduced rank regression in Bayesian estimation, Neural Networks 18 (2005), no. 7, 924–933.
M. S. Bartlett, Contingency table interactions, Supplement to J. Roy. Statist. Soc. 2 (1935), no. 2, 248–252.
V. Baldoni, N. Berline, J. De Loera, B. Dutra, M. Köppe, S. Moreinis, G. Pinto, M. Vergne, and J. Wu, A user’s guide for LattE integrale v1.7.3, available from URL http://www.math.ucdavis.edu/~latte/, 2015.
Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987 French original; Revised by the authors. MR 1659509, DOI 10.1007/978-3-662-03718-8
Joseph Blitzstein and Persi Diaconis, A sequential importance sampling algorithm for generating random graphs with prescribed degrees, Internet Math. 6 (2010), no. 4, 489–522. MR 2809836, DOI 10.1080/15427951.2010.557277
Julian Besag, Spatial interaction and the statistical analysis of lattice systems, J. Roy. Statist. Soc. Ser. B 36 (1974), 192–236. MR 373208
Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler, Bertini: Software for numerical algebraic geometry, available at bertini.nd.edu with permanent doi: dx.doi.org/10.7274/R0H41PB5.
Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler, Numerically solving polynomial systems with Bertini, Software, Environments, and Tools, vol. 25, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. MR 3155500
Patrick Billingsley, Probability and measure, 3rd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1324786
Wayne W. Barrett, Charles R. Johnson, and Raphael Loewy, The real positive definite completion problem: cycle completability, Mem. Amer. Math. Soc. 122 (1996), no. 584, viii+69. MR 1342017, DOI 10.1090/memo/0584
Wayne Barrett, Charles R. Johnson, and Pablo Tarazaga, The real positive definite completion problem for a simple cycle, Linear Algebra Appl. 192 (1993), 3–31. Computational linear algebra in algebraic and related problems (Essen, 1992). MR 1236734, DOI 10.1016/0024-3795(93)90234-F
Daniel Bernstein and Colby Long, L-infinity optimization in tropical geometry and phylogenetics, arXiv:1606.03702, 2016.
Francisco Barahona and Ali Ridha Mahjoub, On the cut polytope, Math. Programming 36 (1986), no. 2, 157–173. MR 866986, DOI 10.1007/BF02592023
Yael Berstein, Hugo Maruri-Aguilar, Shmuel Onn, Eva Riccomagno, and Henry Wynn, Minimal average degree aberration and the state polytope for experimental designs, Ann. Inst. Statist. Math. 62 (2010), no. 4, 673–698. MR 2652311, DOI 10.1007/s10463-010-0291-8
David Bryant, Vincent Moulton, and Andreas Spillner, Neighbornet: an agglomerative method for the construction of planar phylogenetic networks, Mol. Biol. Evol. 21 (2004), 255–265.
Ole Barndorff-Nielsen, Information and exponential families in statistical theory, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1978. MR 489333
Daniel J. Bates and Luke Oeding, Toward a salmon conjecture, Exp. Math. 20 (2011), no. 3, 358–370. MR 2836258, DOI 10.1080/10586458.2011.576539
Brandon Bock, Algebraic and combinatorial properties of statistical models for ranked data, Ph.D. thesis, North Carolina State University, 2015.
R. C. Bose, Mathematical theory of the symmetrical factorial design, Sankhyā 8 (1947), 107–166. MR 26781
Carlos Brito and Judea Pearl, A new identification condition for recursive models with correlated errors, Struct. Equ. Model. 9 (2002), no. 4, 459–474. MR 1930449, DOI 10.1207/S15328007SEM0904_{1}
Saugata Basu, Richard Pollack, and Marie-Françoise Roy, Algorithms in real algebraic geometry, 2nd ed., Algorithms and Computation in Mathematics, vol. 10, Springer-Verlag, Berlin, 2006. MR 2248869
Lawrence D. Brown, Fundamentals of statistical exponential families with applications in statistical decision theory, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 9, Institute of Mathematical Statistics, Hayward, CA, 1986. MR 882001
David Bryant, Extending tree models to splits networks, Algebraic statistics for computational biology, Cambridge Univ. Press, New York, 2005, pp. 322–334. MR 2205882, DOI 10.1017/CBO9780511610684.021
N. Beerenwinkel and S. Sullivant, Markov models for accumulating mutations, Biometrika 96 (2009), no. 3, 645–661. MR 2538763, DOI 10.1093/biomet/asp023
Daniel Irving Bernstein and Seth Sullivant, Normal binary hierarchical models, Exp. Math. 26 (2017), no. 2, 153–164. MR 3623866, DOI 10.1080/10586458.2016.1142911
Grigoriy Blekherman and Rainer Sinn, Maximum likelihood threshold and generic completion rank of graphs, 2017.
G. Bellu, M.P. Saccomani, S. Audoly, and L. D’Angió, Daisy: a new software tool to test global identifiability of biological and physiological systems, Computer Methods and Programs in Biomedicine 88 (2007), 52–61.
Dimitris Bertsimas and John Tsitsiklis, Introduction to linear optimization, 1st ed., Athena Scientific, 1997.
Søren L. Buhl, On the existence of maximum likelihood estimators for graphical Gaussian models, Scand. J. Statist. 20 (1993), no. 3, 263–270. MR 1241392
Peter Buneman, A note on the metric properties of trees, J. Combinatorial Theory Ser. B 17 (1974), 48–50. MR 363963, DOI 10.1016/0095-8956(74)90047-1
J. E. Campbell, On a Law of Combination of Operators (Second Paper), Proc. Lond. Math. Soc. 29 (1897/98), 14–32. MR 1576434, DOI 10.1112/plms/s1-29.1.14
James A. Cavender, Taxonomy with confidence, Math. Biosci. 40 (1978), no. 3-4, 271–280. MR 503936, DOI 10.1016/0025-5564(78)90089-5
Yuguo Chen, Persi Diaconis, Susan P. Holmes, and Jun S. Liu, Sequential Monte Carlo methods for statistical analysis of tables, J. Amer. Statist. Assoc. 100 (2005), no. 469, 109–120. MR 2156822, DOI 10.1198/016214504000001303
Yuguo Chen, Ian H. Dinwoodie, and Seth Sullivant, Sequential importance sampling for multiway tables, Ann. Statist. 34 (2006), no. 1, 523–545. MR 2275252, DOI 10.1214/009053605000000822
James A. Cavender and Joseph Felsenstein, Invariants of phylogenies: a simple case with discrete states, J. Classif. 4 (1987), 57–71.
Victor Chepoi and Bernard Fichet, $l_\infty$-approximation via subdominants, J. Math. Psych. 44 (2000), no. 4, 600–616. MR 1804236, DOI 10.1006/jmps.1999.1270
N. N. Chentsov, Statisticheskie reshayushchie pravila i optimal′nye vyvody, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0343398
Ronald Christensen, Log-linear models and logistic regression, 2nd ed., Springer Texts in Statistics, Springer-Verlag, New York, 1997. MR 1633357
James Cussens, David Haws, and Milan Studený, Polyhedral aspects of score equivalence in Bayesian network structure learning, Math. Program. 164 (2017), no. 1-2, Ser. A, 285–324. MR 3661033, DOI 10.1007/s10107-016-1087-2
Kai Lai Chung, A course in probability theory, 3rd ed., Academic Press, Inc., San Diego, CA, 2001. MR 1796326
Peter Clifford, Markov random fields in statistics, Disorder in physical systems, Oxford Sci. Publ., Oxford Univ. Press, New York, 1990, pp. 19–32. MR 1064553
David A. Cox, John Little, and Donal O’Shea, Using algebraic geometry, 2nd ed., Graduate Texts in Mathematics, vol. 185, Springer, New York, 2005. MR 2122859
David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2007. An introduction to computational algebraic geometry and commutative algebra. MR 2290010, DOI 10.1007/978-0-387-35651-8
Imre Csiszár and František Matúš, Generalized maximum likelihood estimates for exponential families, Probab. Theory Related Fields 141 (2008), no. 1-2, 213–246. MR 2372970, DOI 10.1007/s00440-007-0084-z
Luca Chiantini and Giorgio Ottaviani, On generic identifiability of 3-tensors of small rank, SIAM J. Matrix Anal. Appl. 33 (2012), no. 3, 1018–1037. MR 3023462, DOI 10.1137/110829180
Aldo Conca, Gröbner bases of ideals of minors of a symmetric matrix, J. Algebra 166 (1994), no. 2, 406–421. MR 1279266, DOI 10.1006/jabr.1994.1160
Marta Casanellas and Seth Sullivant, The strand symmetric model, Algebraic statistics for computational biology, Cambridge Univ. Press, New York, 2005, pp. 305–321. MR 2205881, DOI 10.1017/CBO9780511610684.020
James W. Cooley and John W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comp. 19 (1965), 297–301. MR 178586, DOI 10.1090/S0025-5718-1965-0178586-1
Pasqualina Conti and Carlo Traverso, Buchberger algorithm and integer programming, Applied algebra, algebraic algorithms and error-correcting codes (New Orleans, LA, 1991) Lecture Notes in Comput. Sci., vol. 539, Springer, Berlin, 1991, pp. 130–139. MR 1229314, DOI 10.1007/3-540-54522-0_{1}02
Bryant Chen, Jin Tian, and Judea Pearl, Testable implications of linear structural equations models, Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence (AAAI) (Quebec City, Canada), July 27-31, 2014.
William H. E. Day, Computational complexity of inferring phylogenies from dissimilarity matrices, Bull. Math. Biol. 49 (1987), no. 4, 461–467. MR 908160, DOI 10.1016/S0092-8240(87)80007-1
I. R. Dunsmore, F. Daly, and the M345 course team, M345 statistical methods, unit 9: Categorical data, Milton Keynes: The Open University, 1987.
Persi Diaconis and Bradley Efron, Testing for independence in a two-way table: new interpretations of the chi-square statistic, Ann. Statist. 13 (1985), no. 3, 845–913. With discussions and with a reply by the authors. MR 803747, DOI 10.1214/aos/1176349634
Richard M. Durbin, Sean R. Eddy, Anders Krogh, and Graeme Mitchison, Biological sequence analysis: Probabilistic models of proteins and nucleic acids, Cambridge University Press, 1998.
Persi Diaconis, David Eisenbud, and Bernd Sturmfels, Lattice walks and primary decomposition, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996) Progr. Math., vol. 161, Birkhäuser Boston, Boston, MA, 1998, pp. 173–193. MR 1627343
P. Diaconis and D. Freedman, Finite exchangeable sequences, Ann. Probab. 8 (1980), no. 4, 745–764. MR 577313
Adrian Dobra and Stephen E. Fienberg, Bounds for cell entries in contingency tables given marginal totals and decomposable graphs, Proc. Natl. Acad. Sci. USA 97 (2000), no. 22, 11885–11892. MR 1789526, DOI 10.1073/pnas.97.22.11885
Adrian Dobra and Stephen E. Fienberg, Bounds for cell entries in contingency tables induced by fixed marginal totals, UNECE Statistical Journal 18 (2001), 363–371.
Mathias Drton, Rina Foygel, and Seth Sullivant, Global identifiability of linear structural equation models, Ann. Statist. 39 (2011), no. 2, 865–886. MR 2816341, DOI 10.1214/10-AOS859
Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister, and Hans Schönemann, Singular 4-1-0 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de, 2016.
Noah S. Daleo and Jonathan D. Hauenstein, Numerically deciding the arithmetically Cohen-Macaulayness of a projective scheme, J. Symbolic Comput. 72 (2016), 128–146. MR 3369053, DOI 10.1016/j.jsc.2015.01.001
Jan Draisma, Emil Horobeţ, Giorgio Ottaviani, Bernd Sturmfels, and Rekha R. Thomas, The Euclidean distance degree of an algebraic variety, Found. Comput. Math. 16 (2016), no. 1, 99–149. MR 3451425, DOI 10.1007/s10208-014-9240-x
Colin N. Dewey, Peter M. Huggins, Kevin Woods, Bernd Sturmfels, and Lior Pachter, Parametric alignment of Drosophila genomes, PLOS Computational Biology 2 (2006), e73.
Persi Diaconis, Finite forms of de Finetti’s theorem on exchangeability, Synthese 36 (1977), no. 2, 271–281. Foundations of probability and statistics, II. MR 517222, DOI 10.1007/BF00486116
Persi Diaconis, Group representations in probability and statistics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 11, Institute of Mathematical Statistics, Hayward, CA, 1988. MR 964069
G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961), 71–76. MR 130190, DOI 10.1007/BF02992776
Elena S. Dimitrova, Abdul Salam Jarrah, Reinhard Laubenbacher, and Brandilyn Stigler, A Gröbner fan method for biochemical network modeling, ISSAC 2007, ACM, New York, 2007, pp. 122–126. MR 2396193, DOI 10.1145/1277548.1277566
Harm Derksen and Gregor Kemper, Computational invariant theory, Invariant Theory and Algebraic Transformation Groups, I, Springer-Verlag, Berlin, 2002. Encyclopaedia of Mathematical Sciences, 130. MR 1918599, DOI 10.1007/978-3-662-04958-7
Jan Draisma and Jochen Kuttler, On the ideals of equivariant tree models, Math. Ann. 344 (2009), no. 3, 619–644. MR 2501304, DOI 10.1007/s00208-008-0320-6
Michel Marie Deza and Monique Laurent, Geometry of cuts and metrics, Algorithms and Combinatorics, vol. 15, Springer-Verlag, Berlin, 1997. MR 1460488, DOI 10.1007/978-3-642-04295-9
Jesús A. De Loera, Raymond Hemmecke, and Matthias Köppe, Algebraic and geometric ideas in the theory of discrete optimization, MOS-SIAM Series on Optimization, vol. 14, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2013. MR 3024570
Jesús A. De Loera and Shmuel Onn, Markov bases of three-way tables are arbitrarily complicated, J. Symbolic Comput. 41 (2006), no. 2, 173–181. MR 2197153, DOI 10.1016/j.jsc.2005.04.010
A. P. Dempster, N. M. Laird, and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. Roy. Statist. Soc. Ser. B 39 (1977), no. 1, 1–38. With discussion. MR 501537
Mathias Drton, Shaowei Lin, Luca Weihs, and Piotr Zwiernik, Marginal likelihood and model selection for Gaussian latent tree and forest models, Bernoulli 23 (2017), no. 2, 1202–1232. MR 3606764, DOI 10.3150/15-BEJ775
Alicia Dickenstein, Laura Felicia Matusevich, and Ezra Miller, Combinatorics of binomial primary decomposition, Math. Z. 264 (2010), no. 4, 745–763. MR 2593293, DOI 10.1007/s00209-009-0487-x
Adrian Dobra, Statistical tools for disclosure limitation in multi-way contingency tables, ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)–Carnegie Mellon University. MR 2703763
Adrian Dobra, Markov bases for decomposable graphical models, Bernoulli 9 (2003), no. 6, 1093–1108. MR 2046819, DOI 10.3150/bj/1072215202
Mathias Drton and Martyn Plummer, A Bayesian information criterion for singular models, J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 (2017), no. 2, 323–380. With discussions. MR 3611750, DOI 10.1111/rssb.12187
J. N. Darroch and D. Ratcliff, Generalized iterative scaling for log-linear models, Ann. Math. Statist. 43 (1972), 1470–1480. MR 345337, DOI 10.1214/aoms/1177692379
Mathias Drton and Thomas S. Richardson, Graphical methods for efficient likelihood inference in Gaussian covariance models, J. Mach. Learn. Res. 9 (2008), 893–914. MR 2417257
Mathias Drton, Discrete chain graph models, Bernoulli 15 (2009), no. 3, 736–753. MR 2555197, DOI 10.3150/08-BEJ172
Mathias Drton, Likelihood ratio tests and singularities, Ann. Statist. 37 (2009), no. 2, 979–1012. MR 2502658, DOI 10.1214/07-AOS571
Persi Diaconis and Bernd Sturmfels, Algebraic algorithms for sampling from conditional distributions, Ann. Statist. 26 (1998), no. 1, 363–397. MR 1608156, DOI 10.1214/aos/1030563990
Adrian Dobra and Seth Sullivant, A divide-and-conquer algorithm for generating Markov bases of multi-way tables, Comput. Statist. 19 (2004), no. 3, 347–366. MR 2096204, DOI 10.1007/BF03372101
Ruth Davidson and Seth Sullivant, Polyhedral combinatorics of UPGMA cones, Adv. in Appl. Math. 50 (2013), no. 2, 327–338. MR 3003350, DOI 10.1016/j.aam.2012.10.002
Ruth Davidson and Seth Sullivant, Distance-based phylogenetic methods around a polytomy, IEEE/ACM Transactions on Computational Biology and Bioinformatics 11 (2014), 325–335.
Mathias Drton, Bernd Sturmfels, and Seth Sullivant, Algebraic factor analysis: tetrads, pentads and beyond, Probab. Theory Related Fields 138 (2007), no. 3-4, 463–493. MR 2299716, DOI 10.1007/s00440-006-0033-2
Mathias Drton, Bernd Sturmfels, and Seth Sullivant, Lectures on algebraic statistics, Oberwolfach Seminars, vol. 39, Birkhäuser Verlag, Basel, 2009. MR 2723140, DOI 10.1007/978-3-7643-8905-5
Mathias Drton and Luca Weihs, Generic identifiability of linear structural equation models by ancestor decomposition, Scand. J. Stat. 43 (2016), no. 4, 1035–1045. MR 3573674, DOI 10.1111/sjos.12227
Cynthia Dwork, Differential privacy, Automata, languages and programming. Part II, Lecture Notes in Comput. Sci., vol. 4052, Springer, Berlin, 2006, pp. 1–12. MR 2307219, DOI 10.1007/11787006_{1}
Mathias Drton and Josephine Yu, On a parametrization of positive semidefinite matrices with zeros, SIAM J. Matrix Anal. Appl. 31 (2010), no. 5, 2665–2680. MR 2740626, DOI 10.1137/100783170
Robert J. Elliott, Lakhdar Aggoun, and John B. Moore, Hidden Markov models, Applications of Mathematics (New York), vol. 29, Springer-Verlag, New York, 1995. Estimation and control. MR 1323178
Nicholas Eriksson, Stephen E. Fienberg, Alessandro Rinaldo, and Seth Sullivant, Polyhedral conditions for the nonexistence of the MLE for hierarchical log-linear models, J. Symbolic Comput. 41 (2006), no. 2, 222–233. MR 2197157, DOI 10.1016/j.jsc.2005.04.003
Paul Erdos and Tibor Gallai, Gráfok elóírt fokszámú pontokkal, Matematikai Lapok 11 (1960), 264–274.
Michael J. Evans, Zvi Gilula, and Irwin Guttman, Latent class analysis of two-way contingency tables by Bayesian methods, Biometrika 76 (1989), no. 3, 557–563. MR 1040648, DOI 10.1093/biomet/76.3.557
Rob H. Eggermont, Emil Horobeţ, and Kaie Kubjas, Algebraic boundary of matrices of nonnegative rank at most three, Linear Algebra Appl. 508 (2016), 62–80. MR 3542981, DOI 10.1016/j.laa.2016.06.036
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Alexander Engström, Thomas Kahle, and Seth Sullivant, Multigraded commutative algebra of graph decompositions, J. Algebraic Combin. 39 (2014), no. 2, 335–372. MR 3159255, DOI 10.1007/s10801-013-0450-0
Péter L. Erdős, István Miklós, and Zoltán Toroczkai, New classes of degree sequences with fast mixing swap Markov chain sampling, Combin. Probab. Comput. 27 (2018), no. 2, 186–207. MR 3778199, DOI 10.1017/S0963548317000499
Alexander Engström and Patrik Norén, Polytopes from subgraph statistics, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Math. Theor. Comput. Sci. Proc., AO, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2011, pp. 305–316 (English, with English and French summaries). MR 2820719
Alexander Engström, Cut ideals of $K_4$-minor free graphs are generated by quadrics, Michigan Math. J. 60 (2011), no. 3, 705–714. MR 2861096, DOI 10.1307/mmj/1320763056
Nicholas Eriksson, Tree construction using singular value decomposition, Algebraic statistics for computational biology, Cambridge Univ. Press, New York, 2005, pp. 347–358. MR 2205884, DOI 10.1017/CBO9780511610684.023
Steven N. Evans and T. P. Speed, Invariants of some probability models used in phylogenetic inference, Ann. Statist. 21 (1993), no. 1, 355–377. MR 1212181, DOI 10.1214/aos/1176349030
David Eisenbud and Bernd Sturmfels, Binomial ideals, Duke Math. J. 84 (1996), no. 1, 1–45. MR 1394747, DOI 10.1215/S0012-7094-96-08401-X
Sergi Elizalde and Kevin Woods, Bounds on the number of inference functions of a graphical model, Statist. Sinica 17 (2007), no. 4, 1395–1415. MR 2398601
Kord Eickmeyer and Ruriko Yoshida, Geometry of neighbor-joining algorithm for small trees, Proceedings of Algebraic Biology, Springer LNC Series, 2008, pp. 82–96.
James S. Farris, A probability model for inferring evolutionary trees, Syst. Zool. 22 (1973), 250–256.
David Fernández-Baca, Timo Seppäläinen, and Giora Slutzki, Bounds for parametric sequence comparison, Discrete Appl. Math. 118 (2002), no. 3, 181–198. MR 1892967, DOI 10.1016/S0166-218X(01)00206-2
Rina Foygel, Jan Draisma, and Mathias Drton, Half-trek criterion for generic identifiability of linear structural equation models, Ann. Statist. 40 (2012), no. 3, 1682–1713. MR 3015040, DOI 10.1214/12-AOS1012
Joseph Felsenstein, Evolutionary trees from dna sequences: a maximum likelihood approach, J. Molecular Evolution 17 (1981), 368–376.
Joseph Felsenstein, Inferring phylogenies, Sinauer Associates, Inc., Sunderland, 2003.
Shmuel Friedland and Elizabeth Gross, A proof of the set-theoretic version of the salmon conjecture, J. Algebra 356 (2012), 374–379. MR 2891138, DOI 10.1016/j.jalgebra.2012.01.017
Conor Fahey, Serkan Hoşten, Nathan Krieger, and Leslie Timpe, Least squares methods for equidistant tree reconstruction, arXiv:0808.3979, 2008.
Audrey Finkler, Goodness of fit statistics for sparse contingency tables, arXiv:1006.2620, 2010.
Alex Fink, The binomial ideal of the intersection axiom for conditional probabilities, J. Algebraic Combin. 33 (2011), no. 3, 455–463. MR 2772542, DOI 10.1007/s10801-010-0253-5
Franklin M. Fisher, The identification problem in econometrics, McGraw-Hill, 1966.
Stefan Forcey, Logan Keefe, and William Sands, Facets of the balanced minimal evolution polytope, J. Math. Biol. 73 (2016), no. 2, 447–468. MR 3521111, DOI 10.1007/s00285-015-0957-1
Stefan Forcey, Logan Keefe, and William Sands, Split-facets for balanced minimal evolution polytopes and the permutoassociahedron, Bull. Math. Biol. 79 (2017), no. 5, 975–994. MR 3634386, DOI 10.1007/s11538-017-0264-7
Stephen E. Fienberg, Sonja Petrović, and Alessandro Rinaldo, Algebraic statistics for $p_1$ random graph models: Markov bases and their uses, Looking back, Lect. Notes Stat. Proc., vol. 202, Springer, New York, 2011, pp. 21–38. MR 2856692, DOI 10.1007/978-1-4419-9389-2_{2}
Stephen E. Fienberg and Alessandro Rinaldo, Three centuries of categorical data analysis: log-linear models and maximum likelihood estimation, J. Statist. Plann. Inference 137 (2007), no. 11, 3430–3445. MR 2363267, DOI 10.1016/j.jspi.2007.03.022
Stephen E. Fienberg and Aleksandra B. Slavković, A survey of statistical approaches to preserving confidentiality of contingency table entries, Privacy-Preserving Data Mining: Models and Algorithms, Springer, 2008, pp. 291–312.
David Gale, A theorem on flows in networks, Pacific J. Math. 7 (1957), 1073–1082. MR 91855
D. Gusfield, K. Balasubramanian, and D. Naor, Parametric optimization of sequence alignment, Algorithmica 12 (1994), no. 4-5, 312–326. MR 1289485, DOI 10.1007/BF01185430
Robert Grone, Charles R. Johnson, Eduardo M. de Sá, and Henry Wolkowicz, Positive definite completions of partial Hermitian matrices, Linear Algebra Appl. 58 (1984), 109–124. MR 739282, DOI 10.1016/0024-3795(84)90207-6
I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
Dan Geiger, Christopher Meek, and Bernd Sturmfels, On the toric algebra of graphical models, Ann. Statist. 34 (2006), no. 3, 1463–1492. MR 2278364, DOI 10.1214/009053606000000263
Luis David García-Puente, Sonja Petrović, and Seth Sullivant, Graphical models, J. Softw. Algebra Geom. 5 (2013), 1–7. MR 3073716, DOI 10.2140/jsag.2013.5.1
Elizabeth Gross, Sonja Petrović, and Despina Stasi, Goodness of fit for log-linear network models: dynamic Markov bases using hypergraphs, Ann. Inst. Statist. Math. 69 (2017), no. 3, 673–704. MR 3635481, DOI 10.1007/s10463-016-0560-2
Luis David Garcia, Michael Stillman, and Bernd Sturmfels, Algebraic geometry of Bayesian networks, J. Symbolic Comput. 39 (2005), no. 3-4, 331–355. MR 2168286, DOI 10.1016/j.jsc.2004.11.007
Ulf Grenander, Probabilities on algebraic structures, John Wiley & Sons, Inc., New York-London; Almqvist & Wiksell, Stockholm-Göteborg-Uppsala, 1963. MR 0206994
Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at https://faculty.math.illinois.edu/Macaulay2/.
Elizabeth Gross and Seth Sullivant, The maximum likelihood threshold of a graph, Bernoulli 24 (2018), no. 1, 386–407. MR 3706762, DOI 10.3150/16-BEJ881
Sun Wei Guo and Elizabeth A. Thompson, Performing the exact test of Hardy-Weinberg proportions for multiple alleles, Biometrics 48 (1992), 361–372.
L. G. Hačijan, A polynomial algorithm in linear programming, Dokl. Akad. Nauk SSSR 244 (1979), no. 5, 1093–1096 (Russian). MR 522052
G. H. Hardy, Mendelian proportions in a mixed population, Science XXVIII (1908), 49–50.
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1995. A first course; Corrected reprint of the 1992 original. MR 1416564
Brendan Hassett, Introduction to algebraic geometry, Cambridge University Press, Cambridge, 2007. MR 2324354, DOI 10.1017/CBO9780511755224
Hisayuki Hara, Satoshi Aoki, and Akimichi Takemura, Running Markov chain without Markov basis, Harmony of Gröbner bases and the modern industrial society, World Sci. Publ., Hackensack, NJ, 2012, pp. 45–62. MR 2986873, DOI 10.1142/9789814383462_{0}005
Dominique M. A. Haughton, On the choice of a model to fit data from an exponential family, Ann. Statist. 16 (1988), no. 1, 342–355. MR 924875, DOI 10.1214/aos/1176350709
Mark Huber, Yuguo Chen, Ian Dinwoodie, Adrian Dobra, and Mike Nicholas, Monte Carlo algorithms for Hardy-Weinberg proportions, Biometrics 62 (2006), no. 1, 49–53, 314 (English, with English and French summaries). MR 2226555, DOI 10.1111/j.1541-0420.2005.00418.x
Søren Højsgaard, David Edwards, and Steffen Lauritzen, Graphical models with R, Use R!, Springer, New York, 2012. MR 2905395, DOI 10.1007/978-1-4614-2299-0
Raymond Hemmecke and Ralf Hemmecke, 4ti2 - Software for computation of Hilbert bases, Graver bases, toric Gröbner bases, and more, 2003, available at http://www.4ti2.de.
Valerie Hower and Christine E. Heitsch, Parametric analysis of RNA branching configurations, Bull. Math. Biol. 73 (2011), no. 4, 754–776. MR 2785143, DOI 10.1007/s11538-010-9607-3
David C. Haws, Terrell L. Hodge, and Ruriko Yoshida, Optimality of the neighbor joining algorithm and faces of the balanced minimum evolution polytope, Bull. Math. Biol. 73 (2011), no. 11, 2627–2648. MR 2855185, DOI 10.1007/s11538-011-9640-x
Masami Hasegawa, Hirohisa Kishino, and Taka aki Yano, Dating of human-ape splitting by a molecular clock of mitochondrial dna, J. Molecular Evolution 22 (1985), 160–174.
Serkan Hoşten, Amit Khetan, and Bernd Sturmfels, Solving the likelihood equations, Found. Comput. Math. 5 (2005), no. 4, 389–407. MR 2189544, DOI 10.1007/s10208-004-0156-8
Paul W. Holland and Samuel Leinhardt, An exponential family of probability distributions for directed graphs, J. Amer. Statist. Assoc. 76 (1981), no. 373, 33–65. MR 608176
Raymond Hemmecke, Silvia Lindner, and Milan Studený, Characteristic imsets for learning Bayesian network structure, Internat. J. Approx. Reason. 53 (2012), no. 9, 1336–1349. MR 2994270, DOI 10.1016/j.ijar.2012.04.001
Raymond Hemmecke and Kristen A. Nairn, On the Gröbner complexity of matrices, J. Pure Appl. Algebra 213 (2009), no. 8, 1558–1563. MR 2517992, DOI 10.1016/j.jpaa.2008.11.044
Hoon Hong, Alexey Ovchinnikov, Gleb Pogudin, and Chee Yap, Global identifiability of differential models, available at https://cs.nyu.edu/$\sim$pogudin/, 2017.
Mike D. Hendy and David Penny, Complete families of linear invariants for some stochastic models of sequence evolution, with and without molecular clock assumption, J. Comput. Biol. 3 (1996), 19–32.
Serkan Hoşten and Jay Shapiro, Primary decomposition of lattice basis ideals, J. Symbolic Comput. 29 (2000), no. 4-5, 625–639. Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). MR 1769658, DOI 10.1006/jsco.1999.0397
Serkan Hoşten and Seth Sullivant, Gröbner bases and polyhedral geometry of reducible and cyclic models, J. Combin. Theory Ser. A 100 (2002), no. 2, 277–301. MR 1940337, DOI 10.1006/jcta.2002.3301
Serkan Hoşten and Seth Sullivant, Ideals of adjacent minors, J. Algebra 277 (2004), no. 2, 615–642. MR 2067622, DOI 10.1016/j.jalgebra.2004.01.027
Serkan Hoşten and Bernd Sturmfels, Computing the integer programming gap, Combinatorica 27 (2007), no. 3, 367–382. MR 2345814, DOI 10.1007/s00493-007-2057-3
Serkan Hoşten and Seth Sullivant, A finiteness theorem for Markov bases of hierarchical models, J. Combin. Theory Ser. A 114 (2007), no. 2, 311–321. MR 2293094, DOI 10.1016/j.jcta.2006.06.001
Christopher J. Hillar and Seth Sullivant, Finite Gröbner bases in infinite dimensional polynomial rings and applications, Adv. Math. 229 (2012), no. 1, 1–25. MR 2854168, DOI 10.1016/j.aim.2011.08.009
June Huh and Bernd Sturmfels, Likelihood geometry, Combinatorial algebraic geometry, Lecture Notes in Math., vol. 2108, Springer, Cham, 2014, pp. 63–117. MR 3329087, DOI 10.1007/978-3-319-04870-3_{3}
June Huh, The maximum likelihood degree of a very affine variety, Compos. Math. 149 (2013), no. 8, 1245–1266. MR 3103064, DOI 10.1112/S0010437X13007057
June Huh, Varieties with maximum likelihood degree one, J. Algebr. Stat. 5 (2014), no. 1, 1–17. MR 3279951, DOI 10.18409/jas.v5i1.22
Thomas H. Jukes and Charles R. Cantor, Evolution of protein molecules, Mammalian Protein Metabolism, Academic Press, New York, 1969, pp. 21–32.
Anders Nedergaard Jensen, Gfan, a software system for Gröbner fans and tropical varieties, available at http://home.imf.au.dk/jensen/software/gfan/gfan.html.
Anders Nedergaard Jensen, A non-regular Gröbner fan, Discrete Comput. Geom. 37 (2007), no. 3, 443–453. MR 2301528, DOI 10.1007/s00454-006-1289-0
Michael Joswig, Polytope propagation on graphs, Algebraic statistics for computational biology, Cambridge Univ. Press, New York, 2005, pp. 181–192. MR 2205871, DOI 10.1017/CBO9780511610684.010
Thomas Kahle, Decompositions of binomial ideals, Ann. Inst. Statist. Math. 62 (2010), no. 4, 727–745. MR 2652314, DOI 10.1007/s10463-010-0290-9
David Kahle, mpoly: Multivariate polynomials in R, The R Journal 5 (2013), no. 1, 162–170.
Richard M. Karp, Reducibility among combinatorial problems, Complexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972) Plenum, New York, 1972, pp. 85–103. MR 0378476
N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984), no. 4, 373–395. MR 779900, DOI 10.1007/BF02579150
Daphne Koller and Nir Friedman, Probabilistic graphical models, Adaptive Computation and Machine Learning, MIT Press, Cambridge, MA, 2009. Principles and techniques. MR 2778120
David Kahle, Luis Garcia-Puente, and Ruriko Yoshida, algstat: Algebraic statistics in R, 2017, R package version 0.1.1.
David Kahle, Luis Garcia-Puente, and Ruriko Yoshida, latter: LattE and 4ti2 in R, 2017, R package version 0.1.0.
Motoo Kimura, A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences, J. Molecular Evolution 16 (1980), 111–120.
George A. Kirkup, Random variables with completely independent subcollections, J. Algebra 309 (2007), no. 2, 427–454. MR 2303187, DOI 10.1016/j.jalgebra.2006.06.023
Mirko Křivánek and Jaroslav Morávek, NP-hard problems in hierarchical-tree clustering, Acta Inform. 23 (1986), no. 3, 311–323. MR 853580, DOI 10.1007/BF00289116
Thomas Kahle and Ezra Miller, Decompositions of commutative monoid congruences and binomial ideals, Algebra Number Theory 8 (2014), no. 6, 1297–1364. MR 3267140, DOI 10.2140/ant.2014.8.1297
Thomas Kahle, Ezra Miller, and Christopher O’Neill, Irreducible decomposition of binomial ideals, Compos. Math. 152 (2016), no. 6, 1319–1332. MR 3518313, DOI 10.1112/S0010437X16007272
David Kahle, Christopher O’Neill, and Jeff Sommars, A computer algebra system for R: Macaulay2 and the m2r package, arXiv e-prints (2017).
Tamio Koyama, Holonomic modules associated with multivariate normal probabilities of polyhedra, Funkcial. Ekvac. 59 (2016), no. 2, 217–242. MR 3560500, DOI 10.1619/fesi.59.217
Vishesh Karwa, Debdeep Pati, Sonja Petrović, Liam Solus, Nikita Alexeev, Mateja Raič, Dane Wilburne, Robert Williams, and Bowei Yan, Exact tests for stochastic block models, arXiv:1612.06040, 2016.
Thomas Kahle and Johannes Rauh, The markov basis database, http://markov-bases.de/, 2014.
Thomas Kahle, Johannes Rauh, and Seth Sullivant, Positive margins and primary decomposition, J. Commut. Algebra 6 (2014), no. 2, 173–208. MR 3249835, DOI 10.1216/JCA-2014-6-2-173
Kaie Kubjas, Elina Robeva, and Bernd Sturmfels, Fixed points EM algorithm and nonnegative rank boundaries, Ann. Statist. 43 (2015), no. 1, 422–461. MR 3311865, DOI 10.1214/14-AOS1282
Joseph B. Kruskal, More factors than subjects, tests and treatments: an indeterminacy theorem for canonical decomposition and individual differences scaling, Psychometrika 41 (1976), no. 3, 281–293. MR 488592, DOI 10.1007/BF02293554
Joseph B. Kruskal, Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics, Linear Algebra Appl. 18 (1977), no. 2, 95–138. MR 444690, DOI 10.1016/0024-3795(77)90069-6
Tamio Koyama and Akimichi Takemura, Calculation of orthant probabilities by the holonomic gradient method, Jpn. J. Ind. Appl. Math. 32 (2015), no. 1, 187–204. MR 3318908, DOI 10.1007/s13160-015-0166-8
Ravi Kannan, Prasad Tetali, and Santosh Vempala, Simple Markov-chain algorithms for generating bipartite graphs and tournaments, Random Structures Algorithms 14 (1999), no. 4, 293–308. MR 1691976, DOI 10.1002/(SICI)1098-2418(199907)14:4<293::AID-RSA1>3.3.CO;2-7
James A. Lake, A rate-independent technique for analysis of nucleaic acid sequences: evolutionary parsimony, Mol. Biol. Evol. 4 (1987), 167–191.
J. M. Landsberg, Tensors: geometry and applications, Graduate Studies in Mathematics, vol. 128, American Mathematical Society, Providence, RI, 2012. MR 2865915, DOI 10.1090/gsm/128
Steffen L. Lauritzen, Graphical models, Oxford Statistical Science Series, vol. 17, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1419991
Monique Laurent, A tour d’horizon on positive semidefinite and Euclidean distance matrix completion problems, Topics in semidefinite and interior-point methods (Toronto, ON, 1996) Fields Inst. Commun., vol. 18, Amer. Math. Soc., Providence, RI, 1998, pp. 51–76. MR 1607310
Gérard Letac, Lectures on natural exponential families and their variance functions, Monografías de Matemática [Mathematical Monographs], vol. 50, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1992. MR 1182991
Lennart Ljung and Torkel Glad, On global identifiability for arbitrary model parametrizations, Automatica J. IFAC 30 (1994), no. 2, 265–276. MR 1261705, DOI 10.1016/0005-1098(94)90029-9
Shaowei Lin, Ideal-theoretic strategies for asymptotic approximation of marginal likelihood integrals, J. Algebr. Stat. 8 (2017), no. 1, 22–55. MR 3614481, DOI 10.18409/jas.v8i1.47
László Lovász, Large networks and graph limits, American Mathematical Society Colloquium Publications, vol. 60, American Mathematical Society, Providence, RI, 2012. MR 3012035, DOI 10.1090/coll/060
David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2009. With a chapter by James G. Propp and David B. Wilson. MR 2466937, DOI 10.1090/mbk/058
Reinhard Laubenbacher and Brandilyn Stigler, A computational algebra approach to the reverse engineering of gene regulatory networks, J. Theoret. Biol. 229 (2004), no. 4, 523–537. MR 2086931, DOI 10.1016/j.jtbi.2004.04.037
Colby Long and Seth Sullivant, Identifiability of 3-class Jukes-Cantor mixtures, Adv. in Appl. Math. 64 (2015), 89–110. MR 3300329, DOI 10.1016/j.aam.2014.12.003
Shaowei Lin, Bernd Sturmfels, and Zhiqiang Xu, Marginal likelihood integrals for mixtures of independence models, J. Mach. Learn. Res. 10 (2009), 1611–1631. MR 2534873
Diane Maclagan, Antichains of monomial ideals are finite, Proc. Amer. Math. Soc. 129 (2001), no. 6, 1609–1615. MR 1814087, DOI 10.1090/S0002-9939-00-05816-0
Bryan F. J. Manly, Randomization, bootstrap and Monte Carlo methods in biology, 3rd ed., Chapman & Hall/CRC Texts in Statistical Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2007. MR 2257066
Hugo Maruri-Aguilar, Eduardo Sáenz-de-Cabezón, and Henry P. Wynn, Alexander duality in experimental designs, Ann. Inst. Statist. Math. 65 (2013), no. 4, 667–686. MR 3094951, DOI 10.1007/s10463-012-0390-9
František Matúš, On limiting towards the boundaries of exponential families, Kybernetika (Prague) 51 (2015), no. 5, 725–738. MR 3445980, DOI 10.14736/kyb-2015-5-0725
H. M. Möller and B. Buchberger, The construction of multivariate polynomials with preassigned zeros, Computer algebra (Marseille, 1982) Lecture Notes in Comput. Sci., vol. 144, Springer, Berlin-New York, 1982, pp. 24–31. MR 680050
Nicolette Meshkat, Marisa Eisenberg, and Joseph J. DiStefano III, An algorithm for finding globally identifiable parameter combinations of nonlinear ODE models using Gröbner bases, Math. Biosci. 222 (2009), no. 2, 61–72. MR 2584099, DOI 10.1016/j.mbs.2009.08.010
Mateusz Michałek, Geometry of phylogenetic group-based models, J. Algebra 339 (2011), 339–356. MR 2811326, DOI 10.1016/j.jalgebra.2011.05.016
Radu Mihaescu, Dan Levy, and Lior Pachter, Why neighbor-joining works, Algorithmica 54 (2009), no. 1, 1–24. MR 2496663, DOI 10.1007/s00453-007-9116-4
Guido F. Montúfar and Jason Morton, When does a mixture of products contain a product of mixtures?, SIAM J. Discrete Math. 29 (2015), no. 1, 321–347. MR 3310972, DOI 10.1137/140957081
Fatemeh Mohammadi, Divisors on graphs, orientations, syzygies, and system reliability, J. Algebraic Combin. 43 (2016), no. 2, 465–483. MR 3456498, DOI 10.1007/s10801-015-0641-y
Teo Mora and Lorenzo Robbiano, The Gröbner fan of an ideal, J. Symbolic Comput. 6 (1988), no. 2-3, 183–208. Computational aspects of commutative algebra. MR 988412, DOI 10.1016/S0747-7171(88)80042-7
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
Fatemeh Mohammadi, Eduardo Sáenz-de-Cabezón, and Henry P. Wynn, The algebraic method in tree percolation, SIAM J. Discrete Math. 30 (2016), no. 2, 1193–1212. MR 3507549, DOI 10.1137/151003647
Nicolette Meshkat, Seth Sullivant, and Marisa Eisenberg, Identifiability results for several classes of linear compartment models, Bull. Math. Biol. 77 (2015), no. 8, 1620–1651. MR 3421974, DOI 10.1007/s11538-015-0098-0
David Mond, Jim Smith, and Duco van Straten, Stochastic factorizations, sandwiched simplices and the topology of the space of explanations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), no. 2039, 2821–2845. MR 2015992, DOI 10.1098/rspa.2003.1150
Peter G. Norton and Earl V. Dunn, Snoring as a risk factor for disease: an epidemiological survey, British Medical Journal (Clinical Research Ed.) 291 (1985), 630–632.
Jerzy Neyman, Molecular studies of evolution: A source of novel statistical problems, Statistical decision theory and related topics (Proc. Sympos., Purdue Univ., Lafayette, Ind., 1970) Academic Press, New York, 1971, pp. 1–27. MR 0327321
J. R. Norris, Markov chains, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2, Cambridge University Press, Cambridge, 1998. Reprint of 1997 original. MR 1600720
Hidefumi Ohsugi, Normality of cut polytopes of graphs in a minor closed property, Discrete Math. 310 (2010), no. 6-7, 1160–1166. MR 2579848, DOI 10.1016/j.disc.2009.11.012
Mitsunori Ogawa, Hisayuki Hara, and Akimichi Takemura, Graver basis for an undirected graph and its application to testing the beta model of random graphs, Ann. Inst. Statist. Math. 65 (2013), no. 1, 191–212. MR 3011620, DOI 10.1007/s10463-012-0367-8
Shmuel Onn and Bernd Sturmfels, Cutting corners, Adv. in Appl. Math. 23 (1999), no. 1, 29–48. MR 1692984, DOI 10.1006/aama.1999.0645
Yves Pauplin, Direct calculation of a tree length using a distance matrix, J. Molecular Evolution 51 (2000), 41–47.
Karl Pearson, Contributions to the mathematical theory of evolution, Phil. Trans. Roy. Soc. London A 185 (1894), 71–110.
Judea Pearl, Reverend bayes on inference engines: A distributed hierarchical approach, Proceedings of the Second National Conference on Artificial Intelligence. AAAI-82, AAAI Press, Pittsburgh, PA., Menlo Park, California, 1982, pp. 133–136.
Judea Pearl, Causality, 2nd ed., Cambridge University Press, Cambridge, 2009. Models, reasoning, and inference. MR 2548166, DOI 10.1017/CBO9780511803161
Jonas Peters, On the intersection property of conditional independence and its application to causal discovery, J. Causal Inference 3 (2014), 97–108.
Sonja Petrović, A survey of discrete methods in (algebraic) statistics for networks, Algebraic and geometric methods in discrete mathematics, Contemp. Math., vol. 685, Amer. Math. Soc., Providence, RI, 2017, pp. 251–277. MR 3625579, DOI 10.1090/conm/685
Sonja Petrović, Alessandro Rinaldo, and Stephen E. Fienberg, Algebraic statistics for a directed random graph model with reciprocation, Algebraic methods in statistics and probability II, Contemp. Math., vol. 516, Amer. Math. Soc., Providence, RI, 2010, pp. 261–283. MR 2730754, DOI 10.1090/conm/516/10180
Giovanni Pistone, Eva Riccomagno, and Henry P. Wynn, Algebraic statistics, Monographs on Statistics and Applied Probability, vol. 89, Chapman & Hall/CRC, Boca Raton, FL, 2001. Computational commutative algebra in statistics. MR 2332740
Lior Pachter and Bernd Sturmfels, Parametric inference for biological sequence analysis, Proc. Natl. Acad. Sci. USA 101 (2004), no. 46, 16138–16143. MR 2114587, DOI 10.1073/pnas.0406011101
Lior Pachter and Bernd Sturmfels, Tropical geometry of statistical models, Proc. Natl. Acad. Sci. USA 101 (2004), no. 46, 16132–16137. MR 2114586, DOI 10.1073/pnas.0406010101
Lior Pachter and Bernd Sturmfels (eds.), Algebraic statistics for computational biology, Cambridge University Press, New York, 2005. MR 2205865, DOI 10.1017/CBO9780511610684
R Core Team, R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria, 2016.
Claudiu Raicu, Secant varieties of Segre-Veronese varieties, Algebra Number Theory 6 (2012), no. 8, 1817–1868. MR 3033528, DOI 10.2140/ant.2012.6.1817
C. Radhakrishna Rao, Linear statistical inference and its applications, 2nd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-London-Sydney, 1973. MR 0346957
Fabio Rapallo, Markov bases and structural zeros, J. Symbolic Comput. 41 (2006), no. 2, 164–172. MR 2197152, DOI 10.1016/j.jsc.2005.04.002
Alexander A. Razborov, Flag algebras, J. Symbolic Logic 72 (2007), no. 4, 1239–1282. MR 2371204, DOI 10.2178/jsl/1203350785
Christian P. Robert and George Casella, Monte Carlo statistical methods, Springer Texts in Statistics, Springer-Verlag, New York, 1999. MR 1707311, DOI 10.1007/978-1-4757-3071-5
John A. Rhodes, A concise proof of Kruskal’s theorem on tensor decomposition, Linear Algebra Appl. 432 (2010), no. 7, 1818–1824. MR 2592918, DOI 10.1016/j.laa.2009.11.033
Alessandro Rinaldo, Sonja Petrović, and Stephen E. Fienberg, Maximum likelihood estimation in the $\beta$-model, Ann. Statist. 41 (2013), no. 3, 1085–1110. MR 3113804, DOI 10.1214/12-AOS1078
Thomas Richardson and Peter Spirtes, Ancestral graph Markov models, Ann. Statist. 30 (2002), no. 4, 962–1030. MR 1926166, DOI 10.1214/aos/1031689015
John A. Rhodes and Seth Sullivant, Identifiability of large phylogenetic mixture models, Bull. Math. Biol. 74 (2012), no. 1, 212–231. MR 2877216, DOI 10.1007/s11538-011-9672-2
Johannes Rauh and Seth Sullivant, Lifting Markov bases and higher codimension toric fiber products, J. Symbolic Comput. 74 (2016), 276–307. MR 3424043, DOI 10.1016/j.jsc.2015.07.003
H. J. Ryser, Combinatorial properties of matrices of zeros and ones, Canadian J. Math. 9 (1957), 371–377. MR 87622, DOI 10.4153/CJM-1957-044-3
Gideon Schwarz, Estimating the dimension of a model, Ann. Statist. 6 (1978), no. 2, 461–464. MR 468014
P. D. Seymour, Matroids and multicommodity flows, European J. Combin. 2 (1981), no. 3, 257–290. MR 633121, DOI 10.1016/S0195-6698(81)80033-9
Mike Steel and Y. Fu, Classifying and counting linear phylogenetic invariants for the Jukes-Cantor model, J. Comput. Biol. 2 (1995), 39–47.
J. G. Sumner, J. Fernández-Sánchez, and P. D. Jarvis, Lie Markov models, J. Theoret. Biol. 298 (2012), 16–31. MR 2899031, DOI 10.1016/j.jtbi.2011.12.017
Jeremy G. Sumner, Jesús Fernández-Sánchez, Peter D. Jarvis, Bodie T. Kaine, Michael D. Woodhams, and Barbara R. Holland, Is the general time-reversible model bad for molecular phylogenetics?, Syst. Biol. 61 (2012), 1069–1978.
Anna Seigal and Montúfar Guido, Mixtures and products in two graphical models, arXiv:1709.05276, 2017.
Igor R. Shafarevich, Basic algebraic geometry. 1, Translated from the 2007 third Russian edition, Springer, Heidelberg, 2013. Varieties in projective space. MR 3100243
Galen R. Shorack, Probability for statisticians, Springer Texts in Statistics, Springer-Verlag, New York, 2000. MR 1762415
Robert Sokal and Charles Michener, A statistical method for evaluating systematic relationships, University of Kansas Science Bulletin 38 (1958), 1409–1438.
Tomi Silander and Petri Myllymaki, A simple approach for finding the globally optimal bayesian network structure, Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence, AUAI Press, 2006, pp. 445–452.
Christine Sinoquet and Raphaël Mourad (eds.), Probabilistic graphical models for genetics, genomics, and postgenomics, Oxford University Press, Oxford, 2014. MR 3364440, DOI 10.1093/acprof:oso/9780198709022.001.0001
Naruya Saitou and Masatoshi Nei, The neighbor-joining method: a new method for reconstructing phylogenetic trees, Molecular Biology and Evolution 4 (1987), 406–425.
Francisco Santos and Bernd Sturmfels, Higher Lawrence configurations, J. Combin. Theory Ser. A 103 (2003), no. 1, 151–164. MR 1986836, DOI 10.1016/S0097-3165(03)00092-X
Charles Semple and Mike Steel, Phylogenetics, Oxford Lecture Series in Mathematics and its Applications, vol. 24, Oxford University Press, Oxford, 2003. MR 2060009
David Speyer and Bernd Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004), no. 3, 389–411. MR 2071813, DOI 10.1515/advg.2004.023
Bernd Sturmfels and Seth Sullivant, Toric ideals of phylogenetic invariants, J. Comput. Biol. 12 (2005), 204–228.
Bernd Sturmfels and Seth Sullivant, Toric geometry of cuts and splits, Michigan Math. J. 57 (2008), 689–709. Special volume in honor of Melvin Hochster. MR 2492476, DOI 10.1307/mmj/1220879432
Jessica Sidman and Seth Sullivant, Prolongations and computational algebra, Canad. J. Math. 61 (2009), no. 4, 930–949. MR 2541390, DOI 10.4153/CJM-2009-047-5
Despina Stasi, Kayvan Sadeghi, Alessandro Rinaldo, Sonja Petrovic, and Stephen Fienberg, $\beta$ models for random hypergraphs with a given degree sequence, Proceedings of COMPSTAT 2014—21st International Conference on Computational Statistics, Internat. Statist. Inst., The Hague, 2014, pp. 593–600. MR 3372442
Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama, Gröbner deformations of hypergeometric differential equations, Algorithms and Computation in Mathematics, vol. 6, Springer-Verlag, Berlin, 2000. MR 1734566, DOI 10.1007/978-3-662-04112-3
Richard P. Stanley, Enumerative combinatorics. Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR 2868112
Seth Sullivant, Kelli Talaska, and Jan Draisma, Trek separation for Gaussian graphical models, Ann. Statist. 38 (2010), no. 3, 1665–1685. MR 2662356, DOI 10.1214/09-AOS760
Mike Steel, Phylogeny—discrete and random processes in evolution, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 89, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. MR 3601108, DOI 10.1137/1.9781611974485.ch1
Paul D. Stolley, When genius errs: R. A. Fisher and the lung cancer controversy, Am. J. Epidemiol. 133 (1991), 416–425.
Milan Studený, Conditional independence relations have no finite complete characterization, Information Theory, Statistical Decision Functions and Random Processes, 1992, pp. 377–396.
Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949, DOI 10.1090/ulect/008
Milan Studený, Probabilistic conditional independence structures, Information Science and Statistics, Springer, London, 2005. MR 3183760
Bernd Sturmfels and Caroline Uhler, Multivariate Gaussian, semidefinite matrix completion, and convex algebraic geometry, Ann. Inst. Statist. Math. 62 (2010), no. 4, 603–638. MR 2652308, DOI 10.1007/s10463-010-0295-4
Seth Sullivant, Small contingency tables with large gaps, SIAM J. Discrete Math. 18 (2005), no. 4, 787–793. MR 2157826, DOI 10.1137/S0895480104444090
Seth Sullivant, Compressed polytopes and statistical disclosure limitation, Tohoku Math. J. (2) 58 (2006), no. 3, 433–445. MR 2273279
Seth Sullivant, Toric fiber products, J. Algebra 316 (2007), no. 2, 560–577. MR 2356844, DOI 10.1016/j.jalgebra.2006.10.004
Seth Sullivant, Algebraic geometry of Gaussian Bayesian networks, Adv. in Appl. Math. 40 (2008), no. 4, 482–513. MR 2412156, DOI 10.1016/j.aam.2007.04.004
Seth Sullivant, Gaussian conditional independence relations have no finite complete characterization, J. Pure Appl. Algebra 213 (2009), no. 8, 1502–1506. MR 2517987, DOI 10.1016/j.jpaa.2008.11.026
Seth Sullivant, Normal binary graph models, Ann. Inst. Statist. Math. 62 (2010), no. 4, 717–726. MR 2652313, DOI 10.1007/s10463-010-0296-3
Bernd Sturmfels and Piotr Zwiernik, Binary cumulant varieties, Ann. Comb. 17 (2013), no. 1, 229–250. MR 3027579, DOI 10.1007/s00026-012-0174-1
Asya Takken, Monte Carlo goodness-of-fit tests for discrete data, Ph.D. thesis, Stanford University, 1999.
M. Trémolières, I. Combroux, A. Hermann, and P. Nobelis, Conservation status assessment of aquatic habitats within the Rhine floodplain using an index based on macrophytes, Ann. Limnol.-Int. J. Lim. 43 (2007), 233–244.
Rekha R. Thomas, A geometric Buchberger algorithm for integer programming, Math. Oper. Res. 20 (1995), no. 4, 864–884. MR 1378110, DOI 10.1287/moor.20.4.864
Rekha R. Thomas, Lectures in geometric combinatorics, Student Mathematical Library, vol. 33, American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2006. IAS/Park City Mathematical Subseries. MR 2237292, DOI 10.1090/stml/033
Caroline Uhler, Geometry of maximum likelihood estimation in Gaussian graphical models, Ann. Statist. 40 (2012), no. 1, 238–261. MR 3014306, DOI 10.1214/11-AOS957
A. Varchenko, Critical points of the product of powers of linear functions and families of bases of singular vectors, Compositio Math. 97 (1995), no. 3, 385–401. MR 1353281
A. W. van der Vaart, Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 3, Cambridge University Press, Cambridge, 1998. MR 1652247, DOI 10.1017/CBO9780511802256
Cynthia Vinzant, Lower bounds for optimal alignments of binary sequences, Discrete Appl. Math. 157 (2009), no. 15, 3341–3346. MR 2560820, DOI 10.1016/j.dam.2009.06.028
Andrew Viterbi, Error bounds for convolutional codes and an asymptotically optimum decoding algorithm, IEEE Transactions on Information Theory 13 (1967), no. 2, 260–269.
Sumio Watanabe, Algebraic geometry and statistical learning theory, Cambridge Monographs on Applied and Computational Mathematics, vol. 25, Cambridge University Press, Cambridge, 2009. MR 2554932, DOI 10.1017/CBO9780511800474
Joe Whittaker, Graphical models in applied multivariate statistics, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1990. MR 1112133
Tobias Windisch, Rapid mixing and Markov bases, SIAM J. Discrete Math. 30 (2016), no. 4, 2130–2145. MR 3573306, DOI 10.1137/15M1022045
Sewall Wright, The method of path coefficients, Annals of Mathematical Statistics 5 (1934), 161–215.
Jing Xi and Seth Sullivant, Sequential importance sampling for the Ising model, arXiv:1410.4217, 2014.
Jing Xi and Ruriko Yoshida, The characteristic imset polytope of Bayesian networks with ordered nodes, SIAM J. Discrete Math. 29 (2015), no. 2, 697–715. MR 3328137, DOI 10.1137/130933848
Mei Yin, Alessandro Rinaldo, and Sukhada Fadnavis, Asymptotic quantization of exponential random graphs, Ann. Appl. Probab. 26 (2016), no. 6, 3251–3285. MR 3582803, DOI 10.1214/16-AAP1175
Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028, DOI 10.1007/978-1-4613-8431-1
Walter Zucchini and Iain L. MacDonald, Hidden Markov models for time series, Monographs on Statistics and Applied Probability, vol. 110, CRC Press, Boca Raton, FL, 2009. An introduction using R. MR 2523850, DOI 10.1201/9781420010893
Piotr Zwiernik and Jim Q. Smith, Tree cumulants and the geometry of binary tree models, Bernoulli 18 (2012), no. 1, 290–321. MR 2888708, DOI 10.3150/10-BEJ338

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Algebraic Statistics

About this Title

Table of Contents

References [Enhancements On Off] (What's this?)