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This paper considers the model of an arbitrarily distributed signal x observed through an added independent white Gaussian noise w, y=x+w. New relations between the minimal mean-square error of the noncausal estimator and the likelihood ratio between y and w are derived. This is followed by an extended version of a recently derived relation between the mutual information I(x;y) and the minimal mean-square error. These results are applied to derive infinite-dimensional versions of the Fisher information and the de Bruijn identity. A comparison between the causal and noncausal estimation errors yields a restricted form of the logarithmic Sobolev inequality. The derivation of the results is based on the Malliavin calculus

On mutual information, likelihood ratios, and estimation error for the additive Gaussian channel