Index-Based Notation for Random Variable and Probability Space

Hiroki Shibata; Yasufumi Takama

doi:10.20965/jaciii.2019.p0715

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JACIII Vol.23 No.4 pp. 715-718

doi: 10.20965/jaciii.2019.p0715

(2019)

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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

Received:

October 1, 2018

Accepted:

February 12, 2019

Published:

July 20, 2019

Keywords:

probability distribution, random variable, probability space, generative model, mathematical notation

Abstract

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.

Cite this article as:

H. Shibata and Y. Takama, “Index-Based Notation for Random Variable and Probability Space,” J. Adv. Comput. Intell. Intell. Inform., Vol.23 No.4, pp. 715-718, 2019.

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References

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[3] R. Salakhutdinov and A. Mnih, “Bayesian probabilistic matrix factorization using Markov chain Monte Carlo,” 25th Int. Conf. on Machine Learning, pp. 880-887, 2008.
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[12] C. Lin and Y. He, “Joint sentiment/topic model for sentiment analysis,” 18th ACM Conf. on Information and Knowledge Management, pp. 375-384, 2009.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] R. Salakhutdinov and G. Hinton, “Deep Boltzmann machines,” 12th Int. Conf. on Artificial Intelligence and Statistics, pp. 448-445, 2009.

[2] [2] 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.

[3] [3] R. Salakhutdinov and A. Mnih, “Bayesian probabilistic matrix factorization using Markov chain Monte Carlo,” 25th Int. Conf. on Machine Learning, pp. 880-887, 2008.

[4] [4] 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.

[5] [5] A. Einstein, “The foundation of the general theory of relativity,” Annalen der Physik, Vol.49, No.7, pp. 769-822, 1916.

[6] [6] D. E. Smith, “History of mathematics,” Dover Publications, Vol.2, p. 692, 1958.

[7] [7] 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]

[8] [8] Mathematical Society of Japan, “Encyclopedic dictionary of mathematics,” MIT Press, 2nd edition, 3rd printing, 1993.

[9] [9] Y. V. Prohorov and Y. A. Rozanov, “Probability theory,” Springer-Verlag, 1969.

[10] [10] 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.

[11] [11] C. M. Bishop, “Pattern recognition and machine learning,” Springer, 8th printing, 2006.

[12] [12] C. Lin and Y. He, “Joint sentiment/topic model for sentiment analysis,” 18th ACM Conf. on Information and Knowledge Management, pp. 375-384, 2009.

Index-Based Notation for Random Variable and Probability Space

Hiroki Shibata and Yasufumi Takama†

Hiroki Shibata and Yasufumi Takama^†