Tweedie Calculus

Original Abstract

Tweedie's formula is a cornerstone of measurement-error analysis and empirical Bayes. In the Gaussian location model, it recovers posterior means directly from the observed marginal density, bypassing nonparametric deconvolution. Beyond a few classical examples, however, there is no systematic method for determining when such representations exist or how to derive them. This paper develops a general framework for such identities in additive-noise models. I study when posterior functionals admit direct expressions in terms of the observed density -- identities I call \emph{Tweedie representations} -- and show that they are characterized by a linear map, the \emph{Tweedie functional}. Under general conditions, I establish its existence, uniqueness, and continuity. I further show that, in many applications, the Tweedie functional can be expressed as the inverse Fourier transform of an explicit tempered distribution, suitably extended when necessary. This reframes the search for Tweedie-type formulas as a problem in the calculus of tempered distributions. The framework recovers the classical Gaussian case and extends to a broad family of noise distributions for which such representations were previously unavailable. It also goes beyond the standard additive model: in the heteroskedastic Gaussian sequence model, a change of variables restores the required structure conditionally and yields new Tweedie representations.