  We consider mean squared estimation with lookahead of a continuous-time
signal corrupted by additive white Gaussian noise. We show that the mutual
information rate function, i.e., the mutual information rate as function of the
signal-to-noise ratio (SNR), does not, in general, determine the minimum mean
squared error (MMSE) with fixed finite lookahead, in contrast to the special
cases with 0 and infinite lookahead (filtering and smoothing errors),
respectively, which were previously established in the literature. We also
establish a new expectation identity under a generalized observation model
where the Gaussian channel has an SNR jump at $t=0$, capturing the tradeoff
between lookahead and SNR.
  Further, we study the class of continuous-time stationary Gauss-Markov
processes (Ornstein-Uhlenbeck processes) as channel inputs, and explicitly
characterize the behavior of the minimum mean squared error (MMSE) with finite
lookahead and signal-to-noise ratio (SNR). The MMSE with lookahead is shown to
converge exponentially rapidly to the non-causal error, with the exponent being
the reciprocal of the non-causal error. We extend our results to mixtures of
Ornstein-Uhlenbeck processes, and use the insight gained to present lower and
upper bounds on the MMSE with lookahead for a class of stationary Gaussian
input processes, whose spectrum can be expressed as a mixture of
Ornstein-Uhlenbeck spectra.
