A Measurement Theoretical Foundation of Statistics

Author(s)
Shiro Ishikawa

Affiliation(s)

Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan.

Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan.

ABSTRACT

It is a matter of course that Kolmogorov’s probability theory is a very useful mathematical tool for the analysis of statistics. However, this fact never means that statistics is based on Kolmogorov’s probability theory, since it is not guaranteed that mathematics and our world are connected. In order that mathematics asserts some statements concerning our world, a certain theory (so called “world view”) mediates between mathematics and our world. Recently we propose measurement theory (i.e., the theory of the quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics. In this paper, we assert that statistics is based on measurement theory. And, for example, we show, from the pure theoretical point of view (i.e., from the measurement theoretical point of view), that regression analysis can not be justified without Bayes’ theorem. This may imply that even the conventional classification of (Fisher’s) statistics and Bayesian statistics should be reconsidered.

It is a matter of course that Kolmogorov’s probability theory is a very useful mathematical tool for the analysis of statistics. However, this fact never means that statistics is based on Kolmogorov’s probability theory, since it is not guaranteed that mathematics and our world are connected. In order that mathematics asserts some statements concerning our world, a certain theory (so called “world view”) mediates between mathematics and our world. Recently we propose measurement theory (i.e., the theory of the quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics. In this paper, we assert that statistics is based on measurement theory. And, for example, we show, from the pure theoretical point of view (i.e., from the measurement theoretical point of view), that regression analysis can not be justified without Bayes’ theorem. This may imply that even the conventional classification of (Fisher’s) statistics and Bayesian statistics should be reconsidered.

Cite this paper

S. Ishikawa, "A Measurement Theoretical Foundation of Statistics,"*Applied Mathematics*, Vol. 3 No. 3, 2012, pp. 283-292. doi: 10.4236/am.2012.33044.

S. Ishikawa, "A Measurement Theoretical Foundation of Statistics,"

References

[1] A. Kolmogorov, “Foundations of the Theory of Probability (Translation),” 2nd Edition, Chelsea Pub Co., New York, 1960,

[2] S. Ishikawa, “Fuzzy Inferences by Algebraic Method,” Fuzzy Sets and Systems, Vol. 87, No. 2, 1997, pp. 181-200. doi:10.1016/S0165-0114(96)00035-8

[3] S. Ishikawa, “A Quantum Mechanical Approach to Fuzzy Theory,” Fuzzy Sets and Systems, Vol. 90, No. 3, 1997, pp. 277-306. doi:10.1016/S0165-0114(96)00114-5

[4] S. Ishikawa, “Statistics in Measurements,” Fuzzy Sets and Systems, Vol. 116, No. 2, 2000, pp. 141-154. doi:10.1016/S0165-0114(98)00280-2

[5] S. Ishikawa, “Mathematical Foundations of Measurement Theory,” Keio University Press Inc., Tokyo, 2006, pp. 1-335. http://www.keio-up.co.jp/kup/mfomt/

[6] S. Ishikawa, “Fisher’s Method, Bayes’ Method and Kalman Filter in Measurement Theory,” Far East Journal of Theoretical Statistics Vol. 29, No. 1, 2009, pp. 9-23.

[7] S. Ishikawa, “A New Interpretation of Quantum Mechanics,” Journal of quantum information science, Vol. 1, No. 2, 2011, pp. 35-42. doi:10.4236/jqis.2011.12005

[8] S. Ishikawa, “Quantum Mechanics and the Philosophy of Language: Reconsideration of Traditional Philosophies,” Journal of Quantum Information Science, Vol. 2, 2012, pp. 2-9.

[9] G. J. Murphy, “C*-algebras and Operator Theory,” Academic Press, London, 1990.

[10] J. von Neumann, “Mathematical Foundations of Quantum Mechanics,” Springer Verlag, Berlin, 1932.

[11] K. Yosida, “Functional Analysis,” 6th Edition, Springer-Verlag, Berlin, 1980.

[12] E. B. Davies, “Quantum Theory of Open Systems,” Academic Press, London, 1976.

[1] A. Kolmogorov, “Foundations of the Theory of Probability (Translation),” 2nd Edition, Chelsea Pub Co., New York, 1960,

[2] S. Ishikawa, “Fuzzy Inferences by Algebraic Method,” Fuzzy Sets and Systems, Vol. 87, No. 2, 1997, pp. 181-200. doi:10.1016/S0165-0114(96)00035-8

[3] S. Ishikawa, “A Quantum Mechanical Approach to Fuzzy Theory,” Fuzzy Sets and Systems, Vol. 90, No. 3, 1997, pp. 277-306. doi:10.1016/S0165-0114(96)00114-5

[4] S. Ishikawa, “Statistics in Measurements,” Fuzzy Sets and Systems, Vol. 116, No. 2, 2000, pp. 141-154. doi:10.1016/S0165-0114(98)00280-2

[5] S. Ishikawa, “Mathematical Foundations of Measurement Theory,” Keio University Press Inc., Tokyo, 2006, pp. 1-335. http://www.keio-up.co.jp/kup/mfomt/

[6] S. Ishikawa, “Fisher’s Method, Bayes’ Method and Kalman Filter in Measurement Theory,” Far East Journal of Theoretical Statistics Vol. 29, No. 1, 2009, pp. 9-23.

[7] S. Ishikawa, “A New Interpretation of Quantum Mechanics,” Journal of quantum information science, Vol. 1, No. 2, 2011, pp. 35-42. doi:10.4236/jqis.2011.12005

[8] S. Ishikawa, “Quantum Mechanics and the Philosophy of Language: Reconsideration of Traditional Philosophies,” Journal of Quantum Information Science, Vol. 2, 2012, pp. 2-9.

[9] G. J. Murphy, “C*-algebras and Operator Theory,” Academic Press, London, 1990.

[10] J. von Neumann, “Mathematical Foundations of Quantum Mechanics,” Springer Verlag, Berlin, 1932.

[11] K. Yosida, “Functional Analysis,” 6th Edition, Springer-Verlag, Berlin, 1980.

[12] E. B. Davies, “Quantum Theory of Open Systems,” Academic Press, London, 1976.