arXiv 15 Apr 2026 · Mathematics — Statistics Theory
arXiv:2604.14486 · PDF · DOI · OpenAlex · Extracted main text
Tweedie's formula is central to measurement-error analysis and empirical Bayes. Under Gaussian noise, the formula identifies the posterior mean directly from the observed-data density, bypassing nonparametric deconvolution. Beyond a few classical examples, however, no general theory 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 Tweedie representations -- and show that they are governed by a linear map, the Tweedie functional. Under general conditions, I prove that this functional exists, is unique, and is continuous. I also provide a constructive method for deriving it by extending the inverse Fourier transform of an explicit tempered distribution. This recasts the search for Tweedie-type formulas as a problem in the calculus of tempered distributions. The framework recovers the classical Gaussian formula and yields new representations for posterior means under non-Gaussian noise. I apply the method to construct unbiased representations of nonlinear functionals of latent variables and to derive Tweedie formulas for the product-Laplace mechanism used in differential privacy. Finally, I show that the approach extends beyond the standard additive model. In the heteroskedastic Gaussian sequence model, where the noise covariance is itself random, a change of variables restores the required additive-noise structure conditionally, yielding Tweedie representations without additional restrictions on the joint law of the latent parameter and noise covariance.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Raphan, Martin and Simoncelli, Eero P (2011) Least squares estimation without priors or supervision | 0.737 | 3 | 3 | 67% |
| 2 | Efron, Bradley (2016) Empirical Bayes deconvolution estimates | 0.737 | 3 | 2 | 100% |
| 3 | Kammler, David W (2007) A first course in Fourier analysis | 0.644 | 2 | 2 | 100% |
| 4 | Kolmogorov, Andrei Nikolaevich (1950) Unbiased estimates | 0.644 | 2 | 2 | 100% |
| 5 | Schwartz, L (1966) Théorie des distributions | 0.644 | 2 | 2 | 100% |
| 6 | Grafakos, Loukas and others (2008) Classical Fourier analysis | 0.511 | 2 | 2 | 50% |
| 7 | Dwork, Cynthia and Roth, Aaron (2014) The algorithmic foundations of differential privacy | 0.511 | 2 | 1 | 100% |
| 8 | Efron, Bradley (2011) Tweedie’s formula and selection bias | 0.511 | 2 | 1 | 100% |
| 9 | Efron, Bradley (2014) Two modeling strategies for empirical Bayes estimation | 0.511 | 2 | 1 | 100% |
| 10 | Robbins, Herbert (1956) An empirical Bayes approach to statistics | 0.511 | 2 | 1 | 100% |
Showing the top 10 of 77 scored citations.