The Curious Problem of the Normal Inverse Mean: Robustness and Shrinkage
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908625539170304 |
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| author | Ghosh, Soham Chatterjee, Uttaran Datta, Jyotishka |
| author_facet | Ghosh, Soham Chatterjee, Uttaran Datta, Jyotishka |
| contents | In astronomical observations, the estimation of distances from parallaxes is a challenging task due to the inherent measurement errors and the non-linear relationship between the parallax and the distance. This study leverages ideas from robust Bayesian inference to tackle these challenges, investigating a broad class of prior densities for estimating distances with a reduced bias and variance. Through theoretical analysis, simulation experiments, and the application to data from the Gaia Data Release 1 (GDR1), we demonstrate that heavy-tailed priors provide more reliable distance estimates, particularly in the presence of large fractional parallax errors. Theoretical results highlight the "curse of a single observation," where the likelihood dominates the posterior, limiting the impact of the prior. Nevertheless, heavy-tailed priors can delay the explosion of posterior risk, offering a more robust framework for distance estimation. The findings suggest that reciprocal invariant priors, with polynomial decay in their tails, such as the Half-Cauchy and Product Half-Cauchy, are particularly well-suited for this task, providing a balance between bias reduction and variance control. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Curious Problem of the Normal Inverse Mean: Robustness and Shrinkage Ghosh, Soham Chatterjee, Uttaran Datta, Jyotishka Methodology Applications 62C10l, 62F35 In astronomical observations, the estimation of distances from parallaxes is a challenging task due to the inherent measurement errors and the non-linear relationship between the parallax and the distance. This study leverages ideas from robust Bayesian inference to tackle these challenges, investigating a broad class of prior densities for estimating distances with a reduced bias and variance. Through theoretical analysis, simulation experiments, and the application to data from the Gaia Data Release 1 (GDR1), we demonstrate that heavy-tailed priors provide more reliable distance estimates, particularly in the presence of large fractional parallax errors. Theoretical results highlight the "curse of a single observation," where the likelihood dominates the posterior, limiting the impact of the prior. Nevertheless, heavy-tailed priors can delay the explosion of posterior risk, offering a more robust framework for distance estimation. The findings suggest that reciprocal invariant priors, with polynomial decay in their tails, such as the Half-Cauchy and Product Half-Cauchy, are particularly well-suited for this task, providing a balance between bias reduction and variance control. |
| title | The Curious Problem of the Normal Inverse Mean: Robustness and Shrinkage |
| topic | Methodology Applications 62C10l, 62F35 |
| url | https://arxiv.org/abs/2410.20641 |