The relationship between causal and non-causal mismatched estimation in continuous-time AWGN channels

A continuous-time finite-power process with distribution P is observed through an AWGN channel, at a given signal-to-noise ratio (SNR), and is estimated by an estimator that would have minimized the mean-square error if the process had distribution Q. We show that the causal filtering meansquare error (MSE) achieved at SNR level snr is equal to the average value of the noncausal (smoothing) MSE achieved with a channel whose SNR is chosen uniformly distributed between 0 and snr. Emerging as the bridge for equating these two quantities are mutual information and relative entropy. Our result generalizes that of Guo, Shamai and Verdú (2005) from the non-mismatched case, where P = Q, to general P and Q. Among our intermediate results is an extension of Duncan's theorem, that relates mutual information and causal MMSE, to the case of mismatched estimation. Some further extensions and implications are discussed. Key to our findings is the recent result of Verdú on mismatched estimation and relative entropy.