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Discrete statistical models with rational maximum likelihood estimator. (English) Zbl 1465.62184

This paper is a study on the discrete statistical models. The authors show a look at algebraic statistics to build the maximum likelihood estimator (MLE). They present a theorem that characterizes all discrete statistical models whose MLE is a rational function. They examine models with rational MLE such as decomposable graphical models and Bayesian networks. The authors present an algorithm for constructing models with rational MLE, and they discuss its implementation and some experiments. The input is an integer matrix, and the output is a list of models derived from that matrix. Their results suggest that only a very small fraction of Huh’s varieties are statistical models.

MSC:

62R01 Algebraic statistics
62H22 Probabilistic graphical models
14P10 Semialgebraic sets and related spaces

Software:

Macaulay2
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References:

[1] Ay, N., Jost, J., Lê, H.V. and Schwachhöfer, L. (2017). Information Geometry. Cham: Springer. Mathematical Reviews (MathSciNet): MR3701408
· Zbl 1383.53002
[2] Clarke, P. and Cox, D.A. (2020). Moment maps, strict linear precision, and maximum likelihood degree one. Adv. Math. 370 107233. Mathematical Reviews (MathSciNet): MR4103774
Zentralblatt MATH: 07212208
Digital Object Identifier: doi:10.1016/j.aim.2020.107233
· Zbl 1453.14125
[3] Collazo, R.A., Görgen, C. and Smith, J.Q. (2018). Chain Event Graphs. Chapman & Hall/CRC Computer Science and Data Analysis Series. Boca Raton, FL: CRC Press.
[4] Drton, M., Sturmfels, B. and Sullivant, S. (2009). Lectures on Algebraic Statistics. Oberwolfach Seminars 39. Basel: Birkhäuser. · Zbl 1166.13001
[5] Duarte, E. and Görgen, C. (2020). Equations defining probability tree models. J. Symbolic Comput. 99 127-146. · Zbl 1451.13086
[6] Garcia-Puente, L.D. and Sottile, F. (2010). Linear precision for parametric patches. Adv. Comput. Math. 33 191-214. Zentralblatt MATH: 1193.65018
Digital Object Identifier: doi:10.1007/s10444-009-9126-7
· Zbl 1193.65018
[7] Gelfand, I.M., Kapranov, M.M. and Zelevinsky, A.V. (1994). Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Boston, MA: Birkhäuser, Inc. Zentralblatt MATH: 0827.14036
· Zbl 0827.14036
[8] Görgen, C. and Smith, J.Q. (2018). Equivalence classes of staged trees. Bernoulli 24 2676-2692. Zentralblatt MATH: 1419.62124
Digital Object Identifier: doi:10.3150/17-BEJ940
Project Euclid: euclid.bj/1522051221
· Zbl 1419.62124
[9] Grayson, D. and Stillman, M. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.
[10] Huh, J. (2014). Varieties with maximum likelihood degree one. J. Algebr. Stat. 5 1-17. Zentralblatt MATH: 1346.14118
Digital Object Identifier: doi:10.18409/jas.v5i1.22
· Zbl 1346.14118
[11] Huh, J. and Sturmfels, B. (2014). Likelihood geometry. In Combinatorial Algebraic Geometry. Lecture Notes in Math. 2108 63-117. Cham: Springer. Zentralblatt MATH: 1328.14004
· Zbl 1328.14004
[12] Kapranov, M.M. (1991). A characterization of \(A\)-discriminantal hypersurfaces in terms of the logarithmic Gauss map. Math. Ann. 290 277-285. Zentralblatt MATH: 0714.14031
Digital Object Identifier: doi:10.1007/BF01459245
· Zbl 0714.14031
[13] Krasauskas, R. (2002). Toric surface patches. Adv. Comput. Math. 17 89-113. Zentralblatt MATH: 0997.65027
Digital Object Identifier: doi:10.1023/A:1015289823859
· Zbl 0997.65027
[14] Lauritzen, S.L. (1996). Graphical Models. Oxford Statistical Science Series 17. New York: Oxford University Press. · Zbl 0907.62001
[15] Silander, T. and Leong, T.-Y. (2013). A dynamic programming algorithm for learning chain event graphs. In Discovery Science (J. Fürnkranz, E. Hüllermeier and T. Higuchi, eds.). Lecture Notes in Computer Science 8140 201-216. Berlin: Springer.
[16] Smith, J.Q. and Anderson, P.E. (2008). Conditional independence and chain event graphs. Artificial Intelligence 172 42-68. Zentralblatt MATH: 1182.68303
Digital Object Identifier: doi:10.1016/j.artint.2007.05.004
· Zbl 1182.68303
[17] Sullivant, S. (2018). Algebraic Statistics. Graduate Studies in Mathematics 194. Providence, RI: Amer. Math. Soc.
[18] https://github.
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