On Information, Estimation and Lookahead

We consider mean squared estimation with lookahead of a continuous-time signal corrupted by additive white Gaussian noise. We investigate the connections between lookahead in estimation, and information under this model. 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 mean squared error 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. Independently, we define the notion of information utility of finite lookahead, and characterize properties of the same. 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.