If we assume that magnitudes of earthquakes are distributed identically and independently according to a negative-exponential function, then the maximum likelihood estimate proposed by Utsu for the b-value is biased from the true value. We suggest an unbiased alternative estimate which is asymptotically equivalent to the maximum likelihood estimate. The relation between the unbiased estimate and the maximum likelihood estimate are presented from a Bayesian viewpoint. The two estimates are compared in order to show the superiority of the unbiased estimate over the maximum likelihood estimate, on the basis of the expected entropy maximization principle for the predictive distributions. From the same principle, the posterior with the noninformative improper prior is recommended for the inferential distribution of the b-value rather than the standardized likelihood.