Higher order optimal approximation of Csiszar's \(f\)-divergence
Let \((X,\mathcal A,\lambda)\) be a measure space, \(\mu_1,\mu_2\) be probability measures on \((X,\mathcal A)\) absolutely continuous with respect to \(\lambda\), and it is assumed that \(p=d\mu_1/d\lambda\) and \(q=d\mu_2/d\lambda\) are such that \(0<a\leq p/q\leq b<\infty\). For a convex function \(f\), the quantity \[ \Gamma_f(\mu_1,\mu_2)=\int_Xq(x)f\left( \frac{p(x)}{q(x)}\right) \,d\lambda (x) \] is called the \(f\)-divergence (or Csiszár's distance) of \(\mu_1\) and \(\mu_2\) [\textit{I. Csiszár}, Stud. Sci. Math. Hung. 2, 299--318 (1967; Zbl 0157.25802)]. The author obtains several representation formulas for the \(f\)-divergence, which lead, under various assumptions, to precise estimates of \(\Gamma_f\).
scientific article; zbMATH DE number 2167862
scientific article