2000 character limit reached
No eleventh conditional Ingleton inequality (2204.03971v3)
Published 8 Apr 2022 in cs.IT, math.IT, and math.PR
Abstract: A rational probability distribution on four binary random variables $X, Y, Z, U$ is constructed which satisfies the conditional independence relations $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Y]$, $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Z \mid U]$, $[Y \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid Z]$ and $[Z \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid XY]$ but whose entropy vector violates the Ingleton inequality. This settles a recent question of Studen\'y (IEEE Trans. Inf. Theory vol. 67, no. 11) and shows that there are, up to symmetry, precisely ten inclusion-minimal sets of conditional independence assumptions on four discrete random variables which make the Ingleton inequality hold. The last case in the classification of which of these inequalities are essentially conditional is also settled.
- Nigel Boston and Ting-Ting Nan: Large violations of the Ingleton inequality. In 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1588–1593 (2012). doi: 10.1109/Allerton.2012.6483410.
- Tobias Boege, Sonja Petrović and Bernd Sturmfels: Marginal independence models. In Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation, ISSAC ’22, pp. 263–271. Association for Computing Machinery (ACM) (2022). doi: 10.1145/3476446.3536193.
- Terence H. Chan: A combinatorial approach to information inequalities. Commun. Inf. Syst., 1(3):241–253 (2001). doi: 10.4310/CIS.2001.v1.n3.a1.
- Satoru Fujishige: Polymatroidal dependence structure of a set of random variables. Inf. Control, 39(1):55–72 (1978). doi: 10.1016/S0019-9958(78)91063-X.
- Arley Gómez, Carolina Mejía and Juan Andrés Montoya: Defining the almost-entropic regions by algebraic inequalities. Int. J. Inf. Coding Theory, 4(1):1–18 (2017). doi: 10.1504/IJICOT.2017.081456.
- Aubrey W. Ingleton: Representation of matroids. In Dominic J. A. Welsh, ed.: Combinatorial Mathematics and its Applications. Proceedings of a Conference held at the Mathematical Institute, Oxford, from 7–10 July, 1969, pp. 149–167 (1971).
- Tarik Kaced and Andrei Romashchenko: Conditional information inequalities for entropic and almost entropic points. IEEE Trans. Inf. Theory, 59(11):7149–7167 (2013). doi: 10.1109/TIT.2013.2274614.
- František Matúš: Conditional independences among four random variables. II. Combin. Probab. Comput., 4(4):407–417 (1995). doi: 10.1017/S0963548300001747.
- František Matúš: Conditional independence structures examined via minors. Ann. Math. Artif. Intell., 21(1):99–30 (1997). doi: 10.1023/A:1018957117081.
- František Matúš: Conditional independences among four random variables. III. Final conclusion. Combin. Probab. Comput., 8(3):269–276 (1999). doi: 10.1017/S0963548399003740.
- Frantisek Matús: Piecewise linear conditional information inequality. IEEE Trans. Inf. Theory, 52(1):236–238 (2006). doi: 10.1109/TIT.2005.860438.
- František Matúš: Infinitely many information inequalities. In Proceedings of the IEEE ISIT 2007, pp. 41–44 (2007).
- František Matúš: On patterns of conditional independences and covariance signs among binary variables. Acta Math. Hung., 154(2):511–524 (2018). doi: 10.1007/s10474-018-0799-6.
- František Matúš and Lászlo Csirmaz: Entropy region and convolution. IEEE Trans. Inf. Theory, 62(11):6007–6018 (2016). doi: 10.1109/TIT.2016.2601598.
- František Matúš and Milan Studený: Conditional independences among four random variables. I. Combin. Probab. Comput., 4(3):269–278 (1995). doi: 10.1017/S0963548300001644.
- Angus J. Macintyre and Alex J. Wilkie: On the decidability of the real exponential field. In Piergiorgio Odifreddi, ed.: Kreiseliana: About and around Georg Kreisel, pp. 441–467. A. K. Peters (1996).
- Claude E. Shannon: A mathematical theory of communication. Bell Syst. Tech. J., 27:379–423, 623–656 (1948). doi: 10.1002/j.1538-7305.1948.tb01338.x.
- Bernd Sturmfels: Gröbner bases and convex polytopes, vol. 8. American Mathematical Society (AMS) (1996). doi: 10.1090/ulect/008.
- Milan Studený: Conditional independence structures over four discrete random variables revisited: conditional Ingleton inequalities. IEEE Trans. Inf. Theory, 67(11):7030–7049 (2021). doi: 10.1109/TIT.2021.3104250.
- Seth Sullivant: Algebraic Statistics, vol. 194 of Graduate Studies in Mathematics. American Mathematical Society (AMS) (2018). doi: 10.1090/gsm/194.
- Zhen Zhang and Raymond W. Yeung: A non-shannon-type conditional inequality of information quantities. IEEE Trans. Inf. Theory, 43(6):1982–1986 (1997). doi: 10.1109/18.641561.