Editing Empirical Bayes method (section)

=== Gaussian ===
Suppose <math>X, Y</math> are random variables, such that <math>Y</math> is observed, but <math>X</math> is hidden. The problem is to find the expectation of <math>X</math>, conditional on <math>Y</math>. Suppose further that <math>Y|X \sim \mathcal N(X, \Sigma)</math>, that is, <math>Y = X+ Z</math>, where <math>Z</math> is a [[Multivariate normal distribution|multivariate gaussian]] with variance <math>\Sigma</math>.

Then, we have the formula <math display="block">\Sigma \nabla_y \rho(y|x) = \rho(y|x) (x-y)</math>by direct calculation with the probability density function of multivariate gaussians. Integrating over <math>\rho(x)dx</math>, we obtain<math display="block">\Sigma \nabla_y \rho(y) = (\mathbb{E}[x|y] - y) \rho(y) \implies \mathbb{E}[x|y] = y + \Sigma \nabla_y \ln \rho(y)</math>In particular, this means that one can perform Bayesian estimation of <math>X</math> without access to either the prior density of <math>X</math> or the posterior density of <math>Y</math>. The only requirement is to have access to the [[Score (statistics)|score function]] of <math>Y</math>. This has applications in [[Diffusion model#Score-based generative model|score-based generative modeling]].<ref>{{Cite journal |last=Saremi |first=Saeed |last2=Hyvärinen |first2=Aapo |date=2019 |title=Neural Empirical Bayes |url=https://www.jmlr.org/papers/v20/19-216.html |journal=Journal of Machine Learning Research |volume=20 |issue=181 |pages=1–23 |issn=1533-7928}}</ref>