Index-Based Notation for Random Variable and Probability Space
Hiroki Shibata and Yasufumi Takama
Graduate School of System Design, Tokyo Metropolitan University
6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan
In a conventional notation used in many studies, a probability space and state space of random variables is identified by its symbol. However, such a notation makes a formula ambiguous in a large equation. This letter proposes to use an index set to identify the probability space and state space of random variables. It is shown that the proposed notation can increase the generality of formulas without ambiguity.
-  R. Salakhutdinov and G. Hinton, “Deep Boltzmann machines,” 12th Int. Conf. on Artificial Intelligence and Statistics, pp. 448-445, 2009.
-  N. T. Kuong, E. Uchino, and N. Suetake, “IVUS Tissue characterization of coronary plaque by classification restricted Boltzmann machine,” J. Adv. Comput. Intell. Intell. Inform., Vol.21, No.1, pp. 67-73, 2017.
-  R. Salakhutdinov and A. Mnih, “Bayesian probabilistic matrix factorization using Markov chain Monte Carlo,” 25th Int. Conf. on Machine Learning, pp. 880-887, 2008.
-  A. A. Efros and T. K. Leung, “Texture synthesis by non-parametric sampling,” 7th IEEE Int. Conf. on Computer Vision, Vol.2, pp. 1033-1038, 1999.
-  A. Einstein, “The foundation of the general theory of relativity,” Annalen der Physik, Vol.49, No.7, pp. 769-822, 1916.
-  D. E. Smith, “History of mathematics,” Dover Publications, Vol.2, p. 692, 1958.
-  The International Organization for Standardization, “ISO 80000-2:2009(E),” 47 pp., 2009, https://people.engr.ncsu.edu/jwilson/files/mathsigns.pdf [accessed June 17, 2019]
-  Mathematical Society of Japan, “Encyclopedic dictionary of mathematics,” MIT Press, 2nd edition, 3rd printing, 1993.
-  Y. V. Prohorov and Y. A. Rozanov, “Probability theory,” Springer-Verlag, 1969.
-  S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol.PAMI-6, No.6, pp. 724-741, 1984.
-  C. M. Bishop, “Pattern recognition and machine learning,” Springer, 8th printing, 2006.
-  C. Lin and Y. He, “Joint sentiment/topic model for sentiment analysis,” 18th ACM Conf. on Information and Knowledge Management, pp. 375-384, 2009.
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