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In maximum entropy spectral analysis (MESA), one maximizes the integral of , where S(f) is a power spectrum. The resulting spectral estimate, which is equivalent to that obtained by linear prediction and other methods, is popular in speech processing applications. An alternative expression, , is used in optical processing and elsewhere. This paper considers whether the alternative expression leads to spectral estimates useful in speech processing. We investigate the question both theoretically and empirically. The theoretical investigation is based on generalizations of file two estimates-the generalizations take into account prior estimates of the unknown power spectrum. It is shown that both estimates result from applying a generalized version of the principle of maximum entropy, but they differ concerning the quantities that are treated as random variables. The empirical investigation is based on speech synthesized using the different spectral estimates. Although both estimates lead to intelligible speech, speech based on the MESA estimate is qualitatively superior.