Tweedie's Formula

Definition

The Tweedie’s formula provides a way to estimate the conditional expectation of a random variable $\mathbf{x}$ given an observation $\tilde{\mathbf{x}}$ that is related to the $\mathbf{x}$. $E[\mathbf{x}|\tilde{\mathbf{x}}]=\tilde{\mathbf{x}}+\lambda {\nabla }_{\tilde{\mathbf{x}}}\ln p(\tilde{\mathbf{x}})$ where $\tilde{\mathbf{x}}$ is the observed data, $\lambda$ is a scale parameter relates to the variance of the noise, and ${\nabla }_{\tilde{\mathbf{x}}}\ln p(\tilde{\mathbf{x}})$ is the Score Function functioning as a correction term in the formula.