# A General Derivative Identity for the Conditional Mean Estimator in Gaussian Noise and Some Applications

arXiv: https://arxiv.org/abs/2104.01883 · HTML: https://mlsanity.com/paper/2104.01883/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 50 references that are in the map.
- First posted on arXiv 5 Apr 2021.

## Builds on (most cited first, 1 of 1)

- [Estimation in Poisson Noise: Properties of the Conditional Mean Estimator](https://mlsanity.com/paper/1911.03744/) (2019)

## Built on by (most cited first, 9 of 9)

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- [An MMSE Lower Bound via Poincar'e Inequality](https://mlsanity.com/paper/2205.05848/) (2022)
- [Is Memorization Helpful or Harmful? Prior Information Sets the Threshold](https://mlsanity.com/paper/2602.09405/) (2026)
- [Spatially Adaptive Noise Injection](https://mlsanity.com/paper/2609.18466/) (2026)
- [Breakdown of Local Denoising as Semantic Speciation](https://mlsanity.com/paper/2609.38176/) (2026)
- [When Repeated Measurements Hurt Tabular Foundation Models](https://mlsanity.com/paper/2610.16801/) (2026)
- [From Global to Tail Convergence: Dynamics of α-Stable Score-Based Generative Models](https://mlsanity.com/paper/2610.19773/) (2026)

## Abstract (arXiv)

Consider a channel bf Y=bf X+ bf N where bf X is an n-dimensional random vector, and bf N is a Gaussian vector with a covariance matrix bf mathsfKbf N. The object under consideration in this paper is the conditional mean of bf X given bf Y=bf y, that is bf y → E[bf X|bf Y=bf y]. Several identities in the literature connect E[bf X|bf Y=bf 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 is derived. Specifically, for the Markov chain bf U leftrightarrow bf X leftrightarrow bf Y, it is shown that the Jacobian of E[bf U|bf Y=bf y] is given by bf mathsfK_bf N-1 bf Cov ( bf X, bf U | bf Y=bf y). In the second part of the paper, via various choices of bf U, the new identity is used to generalize many of the known identities and derive some new ones. 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. Third, a new connection between the conditional cumulants and the conditional expectation is shown. In particular, it is shown that the k-th derivative of E[X|Y=y] is the (k+1)-th conditional cumulant. The third part of the paper considers some applications. In a first application, the power series and the compositional inverse of E[X|Y=y] are derived. In a second application, the distribution of the estimator error (X-E[X|Y]) is derived. In a third application, we construct consistent estimators (empirical Bayes estimators) of the conditional cumulants from an i.i.d. sequence Y1,...,Yn.