A Divergence Formula for Randomness and Dimension

  • Jack H. Lutz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5635)

Abstract

If S is an infinite sequence over a finite alphabet Σ and β is a probability measure on Σ, then the dimension of S with respect to β, written \(\dim^\beta(S)\), is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension dim(S) when β is the uniform probability measure. This paper shows that dimβ(S) and its dual Dimβ(S), the strong dimension of S with respect to β, can be used in conjunction with randomness to measure the similarity of two probability measures α and β on Σ. Specifically, we prove that the divergence formula

$$ {\mathrm {dim}}^\beta(R) = {\mathrm{Dim}}^\beta(R) =\frac{{\mathcal{H}}(\alpha)}{{\mathcal{H}}(\alpha) + {\mathcal{D}}(\alpha || \beta)} $$

holds whenever α and β are computable, positive probability measures on Σ and R ∈ Σ ∞  is random with respect to α. In this formula, \({\mathcal{H}}(\alpha)\) is the Shannon entropy of α, and \({\mathcal{D}}(\alpha||\beta)\) is the Kullback-Leibler divergence between α and β.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jack H. Lutz
    • 1
  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA

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