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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.14486 |
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| _version_ | 1866913086720442368 |
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| author | Torres, Santiago |
| author_facet | Torres, Santiago |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14486 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tweedie Calculus Torres, Santiago Statistics Theory Econometrics Methodology 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. |
| title | Tweedie Calculus |
| topic | Statistics Theory Econometrics Methodology |
| url | https://arxiv.org/abs/2604.14486 |