Tweedie's forumla

Purpose: Tweedie's formula is used in empirical Bayes estimation, specifically for estimating the posterior mean of a parameter when the prior is unknown or estimated from data.
Application: It is commonly applied when dealing with problems involving noisy data or regularization in a Bayesian setting, especially when recovering a signal from a noisy observation.
Key Idea: The formula allows you to compute the posterior mean using only the marginal distribution of the observed data (without requiring explicit knowledge of the prior). It connects the observed data distribution to the posterior mean.
Common Use Case: Denoising in statistics and signal processing.