## Bernoulli

• Bernoulli
• Volume 27, Number 1 (2021), 135-154.

### Discrete statistical models with rational maximum likelihood estimator

#### Abstract

A discrete statistical model is a subset of a probability simplex. Its maximum likelihood estimator (MLE) is a retraction from that simplex onto the model. We characterize all models for which this retraction is a rational function. This is a contribution via real algebraic geometry which rests on results on Horn uniformization due to Huh and Kapranov. We present an algorithm for constructing models with rational MLE, and we demonstrate it on a range of instances. Our focus lies on models familiar to statisticians, like Bayesian networks, decomposable graphical models and staged trees.

#### Article information

Source
Bernoulli, Volume 27, Number 1 (2021), 135-154.

Dates
Revised: April 2020
First available in Project Euclid: 20 November 2020

https://projecteuclid.org/euclid.bj/1605841239

Digital Object Identifier
doi:10.3150/20-BEJ1231

Mathematical Reviews number (MathSciNet)
MR4177364

Zentralblatt MATH identifier
07282845

#### Citation

Duarte, Eliana; Marigliano, Orlando; Sturmfels, Bernd. Discrete statistical models with rational maximum likelihood estimator. Bernoulli 27 (2021), no. 1, 135--154. doi:10.3150/20-BEJ1231. https://projecteuclid.org/euclid.bj/1605841239

#### References

• [1] Ay, N., Jost, J., Lê, H.V. and Schwachhöfer, L. (2017). Information Geometry. Cham: Springer.
• [2] Clarke, P. and Cox, D.A. (2020). Moment maps, strict linear precision, and maximum likelihood degree one. Adv. Math. 370 107233.
• [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.
• [5] Duarte, E. and Görgen, C. (2020). Equations defining probability tree models. J. Symbolic Comput. 99 127–146.
• [6] Garcia-Puente, L.D. and Sottile, F. (2010). Linear precision for parametric patches. Adv. Comput. Math. 33 191–214.
• [7] Gelfand, I.M., Kapranov, M.M. and Zelevinsky, A.V. (1994). Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Boston, MA: Birkhäuser, Inc.
• [8] Görgen, C. and Smith, J.Q. (2018). Equivalence classes of staged trees. Bernoulli 24 2676–2692.
• [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.
• [11] Huh, J. and Sturmfels, B. (2014). Likelihood geometry. In Combinatorial Algebraic Geometry. Lecture Notes in Math. 2108 63–117. Cham: Springer.
• [12] Kapranov, M.M. (1991). A characterization of $A$-discriminantal hypersurfaces in terms of the logarithmic Gauss map. Math. Ann. 290 277–285.
• [13] Krasauskas, R. (2002). Toric surface patches. Adv. Comput. Math. 17 89–113.
• [14] Lauritzen, S.L. (1996). Graphical Models. Oxford Statistical Science Series 17. New York: Oxford University Press.
• [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.
• [17] Sullivant, S. (2018). Algebraic Statistics. Graduate Studies in Mathematics 194. Providence, RI: Amer. Math. Soc.
• [18] https://github.com/emduart2/DiscreteStatisticalModelsWithRationalMLE.