The maximum entropy ansatz, 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 process in this new
sense, is no longer rational but generated by finitely many fractional-poles.Comment: 18 pages, 3 figure
