Generalized Kullback–Leibler divergence ($D(x,y)=\int [x\log (x/y)-x+y]$)

For nonnegative densities of possibly different total masses, this divergence includes the mass correction $-x+y$ as well as the logarithmic term. It is the Bregman distance of the maximum-entropy regularization functional at a positive reference $y$. Convexity of $s\log s-s$ proves nonnegativity. For equal masses the correction integrates to zero, recovering the Kullback-Leibler divergence; for probability density functions both masses are one. Use the usual extended-value conventions at zero reference density.