Abstract
The gaussoid axioms are conditional independence inference rules which characterize regular Gaussian CI structures over a three-element ground set. It is known that no finite set of inference rules completely describes regular Gaussian CI as the ground set grows. In this article we show that the gaussoid axioms logically imply every inference rule of at most two antecedents which is valid for regular Gaussians over any ground set. The proof is accomplished by exhibiting for each inclusion-minimal gaussoid extension of at most two CI statements a regular Gaussian realization. Moreover we prove that all those gaussoids have rational positive-definite realizations inside every ε-ball around the identity matrix. For the proof we introduce the concept of algebraic Gaussians over arbitrary fields and of positive Gaussians over ordered fields and obtain the same two-antecedental completeness of the gaussoid axioms for algebraic and positive Gaussians over all fields of characteristic zero as a byproduct.
Change history
04 April 2022
A Correction to this paper has been published: https://doi.org/10.1007/s10472-022-09792-4
References
Abo Khamis, M., Kolaitis, P.G., Ngo, H.Q., Suciu, D.: Decision problems in information theory. arXiv:2004.08783 (2020)
Boege, T.: Incidence geometry in the projective plane via almost-principal minors of symmetric matrices. arXiv:2103.02589 (2021)
Boege, T., D’Alì, A., Kahle, T., Sturmfels, B.: The geometry of gaussoids. Found. Comput. Math. 19(4), 775–812 (2019). https://doi.org/10.1007/s10208-018-9396-x
Boege, T., Kahle, T.: Construction methods for gaussoids. Kybernetika 56 (6), 1045–1062 (2020). https://doi.org/10.14736/kyb-2020-6-1045
Bouckaert, R., Hemmecke, R., Lindner, S., Studený, M.: Efficient algorithms for conditional independence inference. J. Mach. Learn. Res. 11, 3453–3479 (2010)
Cox, D.A., Little, J., O’Shea, D.: Ideals, varieties, and algorithms, fourth edn. Undergraduate Texts in Mathematics. Springer, Berlin (2015). https://doi.org/10.1007/978-3-319-16721-3. An introduction to computational algebraic geometry and commutative algebra
Dawid, A.P.: Conditional independence in statistical theory. J. Roy. Statist. Soc. Ser. B 41(1), 1–31 (1979)
Dougherty, R., Freiling, C., Zeger, K.: Linear rank inequalities on five or more variables. arXiv:0910.0284 (2010)
Drton, M., Xiao, H.: Smoothness of gaussian conditional independence models. In: Algebraic Methods in Statistics and Probability II, Contemporary Mathematics. https://doi.org/10.1090/conm/516/10173, vol. 516, pp 155–177. American Mathematical Society (2010)
Geiger, D., Pearl, J.: Logical and algorithmic properties of conditional independence and graphical models. Ann. Stat. 21(4), 2001–2021 (1993). https://doi.org/10.1214/aos/1176349407
Geiger, D., Pearl, J.: Logical and algorithmic properties of conditional independence and graphical models. Ann. Statist. 21(4), 2001–2021 (1993). https://doi.org/10.1214/aos/1176349407
Gong, Z., Aldeen, M., Elsner, L.: A note on a generalized Cramer’s rule. Linear Algebra Appl. 340(1), 253–254 (2002)
Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
Holtz, O., Sturmfels, B.: Hyperdeterminantal relations among symmetric principal minors. J. Algebra 316(2). https://doi.org/10.1016/j.jalgebra.2007.01.039 (2007)
Lauritzen, S.: Graphical models., Oxford Statistical Science series, vol. 17. Oxford University Press, Oxford (1996)
Lauritzen, S., Sadeghi, K.: Unifying Markov properties for graphical models. Ann. Statist. 46(5), 2251–2278 (2018). https://doi.org/10.1214/17-AOS1618
Lněnička, R., Matúš, F.: On Gaussian conditional independence structures. Kybernetika 43(3), 327–342 (2007)
Marshall, M.: Positive polynomials and sums of squares, Mathematical Surveys and monographs, vol. 146 American Mathematical Society (2008)
Matúš, F.: Probabilistic conditional independence structures and matroid theory: background. Int. J. Gen. Syst. 22, 185–196 (1994)
Matúš, F.: Conditional independence structures examined via minors. Ann. Math. Artif. Intell. 21(1), 99–30 (1997). https://doi.org/10.1023/A:1018957117081
Matúš, F.: Towards classification of semigraphoids. Discrete Math. 277(1), 115–145 (2004). https://doi.org/10.1016/S0012-365X(03)00155-9
Matúš, F.: Conditional independences in gaussian vectors and rings of polynomials. In: Kern-Isberner, G., Rödder, W., Kulmann, F. (eds.) Conditionals, Information, and Inference, pp. 152–161. Springer (2005)
Matúš, F.: Conditional independences among four random variables. II. Combin. Probab. Comput. 4(4), 407–417 (1995). https://doi.org/10.1017/S0963548300001747
Matúš, F.: Conditional independences among four random variables. III. Final conclusion. Combin. Probab. Comput. 8(3), 269–276 (1999). https://doi.org/10.1017/S0963548399003740
Matúš, F., Studený, M.: Conditional independences among four random variables. I. Combin. Probab. Comput. 4(3), 269–278 (1995). https://doi.org/10.1017/S0963548300001644
Niepert, M., Gyssens, M., Sayrafi, B., Van Gucht, D.: On the conditional independence implication problem: a lattice-theoretic approach. Artif. Intell. 202, 29–51 (2013). https://doi.org/10.1016/j.artint.2013.06.005
Oxley, J.: Matroid theory, Oxford Graduate Texts in Mathematics, 2nd edn., vol. 21. Oxford University Press, Oxford (2011). https://doi.org/10.1093/acprof:oso/9780198566946.001.0001
Pearl, J.: Probabilistic reasoning in intelligent systems: networks of plausible inference. The Morgan Kaufmann Series in Representation and Reasoning. Morgan Kaufmann, San Mateo CA (1988)
Pearl, J., Paz, A.: GRAPHOIDS: A Graph-based logic for reasoning about relevance relations, or When would x tell you more about y if you already know z. Tech. Rep. CSD-850038 UCLA Computer Science Department (1985)
Šimeček, P.: Classes of Gaussian, discrete and binary representable independence models have no finite characterization. In: Proceedings of Prague Stochastics, vol. 400, pp. 622–631 (2006)
Šimeček, P.: Gaussian representation of independence models over four random variables. In: COMPSTAT Conference (2006)
Šimeček, P.: A short note on discrete representability of independence models. In: Proccedings of the European Workshop on Probabilistic Graphical Models, pp. 287–292 (2006)
Studený, M.: Probabilistic conditional independence structures. Information science and statistics springer (2005)
Studený, M.: Conditional independence and basic markov properties. In: Maathuis, M., Drton, M., Lauritzen, S., Wainwright, M. (eds.) Handbook of graphical models, Chapman & Hall/CRC Handbooks of Modern Statistical Methods, pp. 3–38. CRC Press (2019)
Studený, M.: Conditional independence relations have no finite complete characterization. In: Information Theory, Statistical Decision Functions and Random Processes, Vol. B, pp. 377–396. Kluwer (1992)
Studený, M.: Semigraphoids are two-antecedental approximations of stochastic conditional independence models. In: De Mantaras, R.L., Poole, D., Proceedings, Uncertainty (eds.) , pp 546–552, Morgan Kaufmann (1994)
Sullivant, S.: Gaussian conditional independence relations have no finite complete characterization. J. Pure Appl. Algebra 213(8), 1502–1506 (2009)
Sullivant, S.: Algebraic statistics, Graduate Studies in Mathematics, vol. 194 American Mathematical Society (2018)
Verma, T., Pearl, J.: An algorithm for deciding if a set of observed independencies has a causal explanation. In: Dubois, D., Wellman, M.P., D’Ambrosio, B., Smets, P. (eds.) Uncertainty in Artificial Intelligence. https://doi.org/10.1016/B978-1-4832-8287-9.50049-9, pp 323–330, Morgan Kaufmann (1992)
Acknowledgements
The author wishes to thank Thomas Kahle, Andreas Kretschmer and Milan Studený for helpful discussions, and the anonymous referees for their clear and thoughtful suggestions for improvement of the earlier manuscript. Thanks to Xiangying Chen for spotting an error in the previous version of Example 7.4.
Funding
Open Access funding enabled and organized by Projekt DEAL. This work is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 314838170, GRK 2297 MathCoRe.
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Boege, T. Gaussoids are two-antecedental approximations of Gaussian conditional independence structures. Ann Math Artif Intell 90, 645–673 (2022). https://doi.org/10.1007/s10472-021-09780-0
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DOI: https://doi.org/10.1007/s10472-021-09780-0
Keywords
- Conditional independence
- Inference
- Completeness
- Gaussoid
- Realizability
- Rationality
Mathematics Subject Classification (2010)
- 62B10
- 62R01
- 14P10