The Maximum Entropy Principle in the Absence of a Time-Arrow: Fractional-Pole Models

The maximum entropy (ME) principle, as it is often invoked in the context of time-series analysis, suggests the selection of a power spectrum which is consistent with autocorrelation data and corresponds to a random process least predictable from past observations. We introduce and compare a class of spectra with the property that the underlying random process is least predictable at any given point from the complete set of past and future observations. In this context, randomness is quantified by the size of the corresponding smoothing error and deterministic processes are characterized by integrability of the inverse of their power spectral densities - as opposed to the log-integrability in the classical setting. The power spectrum which is consistent with a partial autocorrelation sequence and corresponds to the most random (MR) process in this new sense, is no longer rational but generated by finitely many fractional-poles.