Maximum entropy and maximum likelihood in spectral estimation

The power spectral measure, an informative feature of a stationary time-discrete stochastic process, describes the relative strength of uncorrelated frequency components that compose the process. In spectral estimation one wants to describe the spectral measures of processes having a prescribed initial block of autocorrelation coefficients. In the continuum of possibilities, two types are frequently distinguished: the one-parameter family of maximum-likelihood spectral measures, so named because they carry the largest possible weight at one specified frequency, and the maximum-entropy spectral measure which is often considered to be the most representative of the entire set of possible solutions. Although these choices are very different, representing, respectively, the most and least predictable of the eligible processes, we show that the maximum-entropy measure is exactly the uniform average of the family of maximum-likelihood measures over its parameter.