In a variety of decision systems, processing is performed not on the underlying signal but on a quantized version. Accordingly, assuming fine quantization, Poor observed a quadratic variation in f-divergences with smooth f. In contrast, we derive a quadratic behavior in the Bayesian probability of error, which corresponds to a nonsmooth f, thereby advancing the state of the art. Unlike Poor's purely variational method, we solve a novel cube-slicing problem, and convert a volume integral to a surface integral in the course of our analysis. In this paper, we elaborate our method, and sharpen our result, a preliminary version of which were outlined in our previous work. © 2009 IEEE.