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In this letter, we show that many existing short-time spectral amplitude (STSA) Bayesian estimators for speech enhancement all have a similarly structured cost function. On this basis, we propose a new cost function that generalizes those of existent Bayesian STSA estimators and then obtain the corresponding closed-form solution for the optimal clean speech STSA. The resulting family of estimators, which we will term the generalized weighted family of STSA estimators (GWSA), features a new parameter that acts only on the estimated clean speech STSA. It is found that this new parameter yields an added flexibility in terms of achievable gain curves when compared to those of existing estimators. Moreover, we show that the new estimator family tends to a Wiener filter for high instantaneous signal-to-noise ratios.