×

Equations defining probability tree models. (English) Zbl 1451.13086

This paper studies staged trees using algebraic geometry. Staged trees are particular graph structures encoding sequences or chains of events that are conditionally independent; the edges on the graph are probability weights. The graphical structure of staged trees together with the probabilistic characteristic on the edges appeals to an algebraic geometric approach to studying these objects. In algebraic geometry, one is interested in studying the geometry of zero sets of systems of polynomials, known as algebraic varieties. This paper studies the systems of equations characterizing classes of staged trees using algebraic geometry to understand certain probabilistic properties and features of algebraic geometric objects associated with staged trees.
Staged trees are similar in spirit to statistical graphical models; both are graphical structures that encode relationships between events in terms of conditional independence, but staged trees are strictly sequential and events (nodes) arise based on the “stages” of these depicted events, whereas events (nodes) in graphical models are random variables. Also, the conditional independence in staged trees is between the stages, while in graphical models, it arises in the edges between the random events. Graphical models have been studied using algebraic geometry; in particular, it has been shown that decomposable graphical models can be characterized as toric varieties [G. Pistone et al., Algebraic statistics: Computational commutative algebra in statistics. Boca Raton, FL: Chapman & Hall (2001); Zbl 0960.62003; D. Geiger et al., Ann. Stat. 34, No. 3, 1463–1492 (2006; Zbl 1104.60007)]. Toric varieties are important objects in algebraic geometry: they are algebraic varieties that contain algebraic tori, where algebraic actions on the torus extend to the entire variety. In other words, they encode a local-to-global property, locally from the torus, to the entire variety, globally. Such local-to-global characterizations are of great interest in statistical inference frameworks. A local-to-global property is desirable for staged trees, because it would provide insight on the full sequence of events on the entire staged tree based on studying local observations of events (stages). Staged trees, in general, do not correspond to toric varieties, so an algebraic characterization of such a local-to-global property is missing for staged trees. This paper fills in this gap and gives conditions under which staged trees can be described by toric varieties.
This paper also provides other algebraic geometric properties of this toric variety of staged trees. In particular, it studies the toric ideal of staged trees using its combinatorial structure and gives its generators.

MSC:

13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
62R01 Algebraic statistics
62H22 Probabilistic graphical models
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Allman, E. S.; Rhodes, J. A., Phylogenetic ideals and varieties for the general Markov model, Adv. Appl. Math., 40, 2, 127-148 (2008) · Zbl 1131.92046
[2] Altham, P. M.E., Exact Bayesian analysis of a \(2 \times 2\) contingency table, and Fisher’s “exact” significance test, J. R. Stat. Soc. Ser. B, 31, 261-269 (1969)
[3] Altham, P. M.E., The measurement of association in a contingency table: three extensions of the cross-ratios and metrics methods, J. R. Stat. Soc. Ser. B, 31, 395-407 (1970) · Zbl 0238.62021
[4] Altham, P. M.E., The measurement of association of rows and columns for an \(r \times s\) contingency table, J. R. Stat. Soc. Ser. B, 32, 63-73 (1970) · Zbl 0204.52903
[5] Barclay, L. M.; Collazo, R. A.; Smith, J. Q.; Thwaites, P.; Nicholson, A., The dynamic chain event graph, Electron. J. Stat., 9, 2, 2130-2169 (2015) · Zbl 1336.62205
[6] Barclay, L. M.; Hutton, J. L.; Smith, J. Q., Refining a Bayesian Network using a Chain Event Graph, Int. J. Approx. Reason., 54, 9, 1300-1309 (2013) · Zbl 1316.68169
[7] Boutilier, C.; Friedman, N.; Goldszmidt, M.; Koller, D., Context-specific independence in Bayesian networks, (Horvitz, E.; Jensen, F., 12th Conference on Uncertainty in Artificial Intelligence (UAI 96). Uncertainty in Artificial Intelligence (1996), Morgan Kaufmann Publishers Inc.: Morgan Kaufmann Publishers Inc. San Francisco), 115-123
[8] Casanellas, M.; Fernández-Sánchez, J., Geometry of the Kimura 3-parameter model, Adv. Appl. Math., 41, 3, 265-292 (2008) · Zbl 1222.14110
[9] Collazo, R. A.; Görgen, C.; Smith, J. Q., Chain Event Graphs, Computer Science and Data Analysis Series (2018), Chapman & Hall · Zbl 1391.05001
[10] Collazo, R. A.; Smith, J. Q., A new family of non-local priors for chain event graph model selection, Bayesian Anal., 11, 4, 1165-1201 (2015) · Zbl 1357.62110
[11] Cox, D.; Little, J.; O’Shea, D., Ideals, Varieties, and Algorithms, Vol. 3 (2007), Springer · Zbl 1118.13001
[12] Cox, D. A.; Little, J. B.; Schenck, H. K., Toric Varieties, Graduate Studies in Mathematics, vol. 124 (2011), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1223.14001
[13] Drton, M.; Sturmfels, B.; Sullivant, S., Lectures on Algebraic Statistics, Oberwolfach Seminars, vol. 39 (2009), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1166.13001
[14] Freeman, G.; Smith, J. Q., Dynamic staged trees for discrete multivariate time series: forecasting, model selection and causal analysis, Bayesian Anal., 6, 2, 279-305 (2011) · Zbl 1330.62337
[15] Garcia Puente, L.D., Petrovic, S., Stillman, M., Sullivant, S., 2005a. GraphicalModels: A package for discrete and Gaussian graphical models. Version 1.0.
[16] Garcia Puente, L. D.; Stillman, M.; Sturmfels, B., Algebraic geometry of Bayesian networks, J. Symb. Comput., 39, 3-4, 331-355 (2005) · Zbl 1126.68102
[17] Garthwaite, P. H.; Kadane, J. B.; O’Hagan, A., Statistical method for eliciting probability distributions, J. Am. Stat. Assoc., 100, 470, 680-700 (2005) · Zbl 1117.62340
[18] Geiger, D.; Heckerman, D.; King, H.; Meek, C., Stratified exponential families: graphical models and model selection, Ann. Stat., 29, 2, 505-529 (2001) · Zbl 1012.62012
[19] Geiger, D.; Meek, C.; Sturmfels, B., On the toric algebra of graphical models, Ann. Stat., 34, 3, 1463-1492 (2006) · Zbl 1104.60007
[20] Görgen, C., An Algebraic Characterisation of Staged Trees: Their Geometry and Causal Implications (2017), University of Warwick, Department of Statistics, Ph.D. thesis
[21] Görgen, C.; Bigatti, A.; Riccomagno, E.; Smith, J. Q., Discovery of statistical equivalence classes using computer algebra, Int. J. Approx. Reason., 95, 167-184 (2018) · Zbl 1451.62166
[22] Görgen, C.; Smith, J. Q., Equivalence classes of staged trees, Bernoulli, 24, 4A, 2676-2692 (2018) · Zbl 1419.62124
[23] Grayson, D. R.; Stillman, M. E., Macaulay2, a software system for research in algebraic geometry (2017), Available at
[24] Lauritzen, S. L., Graphical Models, Oxford Statistical Science Series, vol. 17 (1996), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York, Oxford Science Publications · Zbl 0907.62001
[25] Miller, E.; Sturmfels, B., Combinatorial Commutative Algebra, Vol. 227 (2004), Springer Science & Business Media
[26] Pistone, G.; Riccomagno, E.; Wynn, H. P., Algebraic Statistics: Computational Commutative Algebra in Statistics, Monographs on Statistics and Applied Probability, vol. 89 (2001), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL · Zbl 0960.62003
[27] Shafer, G., The Art of Causal Conjecture. Artificial Management (1996), MIT Press: MIT Press Cambridge · Zbl 0874.60003
[28] Smith, J. Q.; Anderson, P. E., Conditional independence and chain event graphs, Artif. Intell., 172, 1, 42-68 (2008) · Zbl 1182.68303
[29] Sullivant, S., Algebraic Statistics, Graduate Studies in Mathematics (2018), American Mathematical Society · Zbl 1408.62004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.