Abstract
Let X and Y be discrete random variables having probability distributions P X and P Y , respectively. A necessary and sufficient condition is obtained for the existence of an α-coupling of these random variables, i.e., for the existence of their joint distribution such that Pr{X = Y} = α, where α, 0 ≤ α ≤ 1, is a given constant. This problem is closely related with the problem of determining the minima of the divergences D(P Z ‖ P X ) and D(P X ‖ P Z ) over all probability distributions P Z of a random variable Z given P X and under the condition that Pr{Z = X} = α. An explicit solution for this problem is also obtained.
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References
Zhang, Z., Estimating Mutual Information via Kolmogorov Distance, IEEE Trans. Inform. Theory, 2007, vol. 53, no. 9, pp. 3280–3282.
Sason, I., Entropy Bounds for Discrete Random Variables via Maximal Coupling, IEEE Trans. Inform. Theory, 2013, vol. 59, no. 11, pp. 7118–7131.
Strassen, V., The Existence of Probability Measures with Given Marginals, Ann. Math. Statist., 1965, vol. 36, no. 2, pp. 423–439.
Marton, K., A Simple Proof of the Blowing-Up Lemma, IEEE Trans. Inform. Theory, 1986, vol. 32, no. 3, pp. 445–446.
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Original Russian Text © V.V. Prelov, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, ^no. 2, pp. 114–121.
The research was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50-00150.
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Prelov, V.V. Coupling of probability distributions and an extremal problem for the divergence. Probl Inf Transm 51, 192–199 (2015). https://doi.org/10.1134/S003294601502009X
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DOI: https://doi.org/10.1134/S003294601502009X