Tweedie Calculus

Abstract

Tweedie's formula, which under Gaussian noise expresses the posterior mean of a latent variable directly from the observed-data density, is a cornerstone of empirical Bayes and measurement-error analysis. No general theory, however, explains when analogous identities hold, how they are structured, or how to derive them for non-Gaussian noise and for posterior functionals other than the mean. This paper develops such a framework for additive-noise models. I characterize when conditional expectations of an unobserved latent variable, given the observed signal, admit direct expressions in terms of the observed density -- identities I call \emph{Tweedie representations} -- and show that they are governed by a linear map, the \emph{Tweedie functional}. Under general conditions, I prove that this functional exists, is unique, and is continuous. I provide a constructive method for its computation based on Fourier analysis: the functional is obtained by extending the inverse Fourier transform of an explicit tempered distribution. The theory yields posterior-mean formulas for non-Gaussian noise and provides new representations for nonlinear posterior functionals. Applications include Laplace mechanisms in differential privacy and heteroskedastic Gaussian sequence models in compound decision problems.