Abstract
The concepts of uncertainty in prediction and inference are introduced and illustrated using the diffraction of light as an example. The close relationship between the concepts of uncertainty in inference and resolving power is noted. A general quantitative measure of uncertainty in inference can be obtained by means of the so-called statistical distance between probability distributions. When applied to quantum mechanics, this distance leads to a measure of the distinguishability of quantum states, which essentially is the absolute value of the matrix element between the states. The importance of this result to the quantum mechanical uncertainty principle is noted. The second part of the paper provides a derivation of the statistical distance on the basis of the so-called method of support.
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References
Jan Hilgevoord and Jos Uffink, inSixty-two Years of Uncertainty: Historical, Philosophical, and Physics Inquiries into the Foundations of Quantum Mechanics, A. I. Miller, ed. (Plenum, New York, 1990).
Jos Uffink,Measures of Uncertainty and the Uncertainty Principle (University of Utrecht thesis, 1990).
W. K. Wootters,Phys. Rev. D 23, 357 (1981).
Jos Uffink and Jan Hilgevoord,Found. Phys. 15, 925 (1985).
V. Barnett,Comparative Statistical Inference, 2nd edn. (Wiley, New York, 1982).
I. Hacking,Logic of Statistical Inference (Cambridge University Press, Cambridge, 1965).
A. W. F. Edwards,Likelihood (Cambridge University Press, Cambridge, 1972).
H. Jeffreys,Proc. Cambridge Philos. Soc. 32, 618 (1936).
H. Jeffreys,Theory of Probability, 2nd edn. (Clarendon Press, Oxford, 1948).
I. J. Good,Probability and the Weighing of Evidence (Charles Griffin, London, 1950).
S. Kullback,Information Theory and Statistics (Wiley, New York, 1959).
E. J. G. Pitman,Some Basic Theory for Statistical Inference (Chapman & Hall, London, 1979).
M. G. Kendall and A. Stuart,The Advanced Theory of Statistics, Vol. 2, 3rd edn. (Charles Griffin, London, 1973), p. 9.
C. R. Rao,Bull. Calcutta Math. Soc. 37, 81 (1945).
See Ref. 9, p. 157.
See Ref. 9, p. 143.
A. Bhattacharyya,Bull. Calcutta Math. Soc. 35, 99 (1943).
S. Kakutani,Ann. Math. 49, 214 (1948).
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Hilgevoord, J., Uffink, J. Uncertainty in prediction and in inference. Found Phys 21, 323–341 (1991). https://doi.org/10.1007/BF01883638
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DOI: https://doi.org/10.1007/BF01883638