Some general divergence measures for probability distributions (Q2368582)

Let \((X,\mathcal{A})\) be a measurable space, \(\mu\) a \(\sigma\)-finite measure on \((X,\mathcal{A})\), \(P\) and \(Q\) probability measures on \((X,\mathcal{A})\) absolutely continuous with respect to \(\mu\). Denote, respectively, by \(p\) and \(q\) the Radon-Nikodým derivatives of \(P\) and \(Q\) with respect to \(\mu\). Let \(f:[0,+\infty)\longrightarrow(-\infty,+\infty]\) be a convex function which is continuous at \(0\). The \(f\)-divergence of \(Q\) and \(P\) is defined by \[ I_f(Q,P)= \int_Xp(x)f \left(\frac{q(x)}{p(x)}\right)\,d\mu(x). \] The author considers more general divergence measures and investigates their properties. Connections with the \(f\)-divergence are explored.