Conditional Mean Estimation in Gaussian Noise: A Meta Derivative Identity With Applications

Consider a channelY= X+ NwhereXis ann-dimensional random vector, andNis a multivariate Gaussian vector with a full-rank covariance matrixKN. The object under consideration in this paper is the conditional mean ofXgivenY=y, that isy ↦ E [X| Y=y]. Several identities in the literature connectE[X| Y=y]to other quantities such as the conditional variance, score functions, and higher-order conditional moments. The objective of this paper is to provide a unifying view of these identities. In the first part of the paper, a general derivative identity for the conditional mean estimator is derived. Specifically, for the Markov chainU↔ X↔ Y, it is shown that the Jacobian matrix ofE[U| Y=y]is given byKN⁻¹ Cov (X, U| Y=y)whereCov (X, U| Y=y)is the conditional covariance. In the second part of the paper, via various choices of the random vectorU, the new identity is used to recover and generalize many of the known identities and derive some new identities. First, a simple proof of the Hatsel and Nolte identity for the conditional variance is shown. Second, a simple proof of the recursive identity due to Jaffer is provided. The Jaffer identity is then further explored, and several equivalent statements are derived, such as an identity for the higher-order conditional expectation (i.e.,E[Xᵏ| Y]) in terms of the derivatives of the conditional expectation. Third, a new fundamental connection between the conditional cumulants and the conditional expectation is demonstrated. In particular, in the univariate case, it is shown that thek-th derivative of the conditional expectation is proportional to the$(k+1)$-th conditional cumulant. A similar expression is derived in the multivariate case.