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It is known that the maximum entropy stationary Gaussian stochastic process, subject to a finite number of autocorrelation constraints, is the Gauss-Markov process of appropriate order. The associated spectrum is Burg's maximum entropy spectral density. We pose a somewhat broader entropy maximization problem, in which stationarity and normality are not assumed, and shift the burden of proof from the previous focus on the calculus of variations and time series techniques to a string of information-theoretic inequalities. This results in an elementary proof of greater generality.