A divergence formula for randomness and dimension (Q616503)

The reviewer considers the first part of the author's abstract as most illustrative of the content of the paper. ``If \(S\) is an infinite sequence over a finite alphabet \(\Sigma\) and \(\beta\) is a probability measure on \(\Sigma\), then the \textit{dimension} of \(S\) with respect to \(\beta\), written \(\mathrm{dim}^\beta(S)\), is a constructive version of the Billingsley dimension that coincides with the (constructive Hausdorff) dimension \(\mathrm{dim}(S)\) when \(\beta\) is the uniform probability measure. This paper shows that \(\mathrm{dim}^\beta(S)\) and its dual \(\mathrm{Dim}^\beta(S)\), the \textit{strong dimension} of \(S\) with respect to \(\beta\), can be used in conjunction with randomness to measure the similarity of two probability measures \(\alpha\) and \(\beta\) on \(\Sigma\). 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 \(\alpha\) and \(\beta\) are computable, positive probability measures on \(\Sigma\) and \(R\in \Sigma^\infty\) is random with respect to \(\alpha\). In this formula, \(\mathcal H (\alpha)\) is the Shannon entropy of \(\alpha\), and \(\mathcal D(\alpha\, \|\, \beta)\) is the Kullback-Leibler divergence between \(\alpha\) and \(\beta\).''