  Many of the classical and recent relations between information and estimation
in the presence of Gaussian noise can be viewed as identities between
expectations of random quantities. These include the I-MMSE relationship of Guo
et al.; the relative entropy and mismatched estimation relationship of
Verd\'{u}; the relationship between causal estimation and mutual information of
Duncan, and its extension to the presence of feedback by Kadota et al.; the
relationship between causal and non-casual estimation of Guo et al., and its
mismatched version of Weissman. We dispense with the expectations and explore
the nature of the pointwise relations between the respective random quantities.
The pointwise relations that we find are as succinctly stated as - and give
considerable insight into - the original expectation identities.
  As an illustration of our results, consider Duncan's 1970 discovery that the
mutual information is equal to the causal MMSE in the AWGN channel, which can
equivalently be expressed saying that the difference between the input-output
information density and half the causal estimation error is a zero mean random
variable (regardless of the distribution of the channel input). We characterize
this random variable explicitly, rather than merely its expectation. Classical
estimation and information theoretic quantities emerge with new and surprising
roles. For example, the variance of this random variable turns out to be given
by the causal MMSE (which, in turn, is equal to the mutual information by
Duncan's result).
