The inverse relation between mutual information (MI) and Bayesian error is sharpened by deriving finite sequences of upper and lower bounds on MI in terms of the minimum probability of error (MPE) and related Bayesian quantities. The well known Fano upper bound and Feder-Merhav lower bound on equivocation are tightened by including a succession of posterior probabilities starting at the largest, which directly controls the MPE, and proceeding to successively lower ones. A number of other interesting results are also derived, including a sequence of upper bounds on the MPE in terms of a previously introduced sequence of generalized posterior distributions. The tightness of the various bounds is illustrated for a simple application of joint spatial localization and spectral typing of a point source.