Multivariate Priors and the Linearity of Optimal Bayesian Estimators under Gaussian Noise
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910311239385088 |
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| author | Barnes, Leighton P. Dytso, Alex Liu, Jingbo Poor, H. Vincent |
| author_facet | Barnes, Leighton P. Dytso, Alex Liu, Jingbo Poor, H. Vincent |
| contents | Consider the task of estimating a random vector $X$ from noisy observations $Y = X + Z$, where $Z$ is a standard normal vector, under the $L^p$ fidelity criterion. This work establishes that, for $1 \leq p \leq 2$, the optimal Bayesian estimator is linear and positive definite if and only if the prior distribution on $X$ is a (non-degenerate) multivariate Gaussian. Furthermore, for $p > 2$, it is demonstrated that there are infinitely many priors that can induce such an estimator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16701 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multivariate Priors and the Linearity of Optimal Bayesian Estimators under Gaussian Noise Barnes, Leighton P. Dytso, Alex Liu, Jingbo Poor, H. Vincent Statistics Theory Information Theory Consider the task of estimating a random vector $X$ from noisy observations $Y = X + Z$, where $Z$ is a standard normal vector, under the $L^p$ fidelity criterion. This work establishes that, for $1 \leq p \leq 2$, the optimal Bayesian estimator is linear and positive definite if and only if the prior distribution on $X$ is a (non-degenerate) multivariate Gaussian. Furthermore, for $p > 2$, it is demonstrated that there are infinitely many priors that can induce such an estimator. |
| title | Multivariate Priors and the Linearity of Optimal Bayesian Estimators under Gaussian Noise |
| topic | Statistics Theory Information Theory |
| url | https://arxiv.org/abs/2401.16701 |