Scale estimation and rate-unbiasedness for Gaussian processes under smoothness misspecification
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866908612720328704 |
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| author | Karvonen, Toni Bachoc, François |
| author_facet | Karvonen, Toni Bachoc, François |
| contents | Gaussian process regression is used throughout statistics and machine learning for prediction and uncertainty quantification. A Gaussian process is specified by its mean and covariance functions. Many covariance functions, including Matérns, have a smoothness parameter that is notoriously difficult to specify correctly or estimate from the data. In practice, the smoothness parameter is often selected more or less arbitrarily. We introduce rate-unbiasedness, a relaxed notion of asymptotic optimality which requires that the expected ratio of the mean-square error presumed by a potentially misspecified model and the true, but unknown, mean-square error remain bounded away from zero and infinity as more data are obtained. A rate-unbiased model provides uncertainty quantification that is of correct order of magnitude. We then prove that scale estimation suffices for rate-unbiasedness in a variety of common settings. As estimation of the scale of a Gaussian process is routine and requires no optimisation, rate-unbiasedness can be achieved in many applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_02810 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Scale estimation and rate-unbiasedness for Gaussian processes under smoothness misspecification Karvonen, Toni Bachoc, François Statistics Theory Gaussian process regression is used throughout statistics and machine learning for prediction and uncertainty quantification. A Gaussian process is specified by its mean and covariance functions. Many covariance functions, including Matérns, have a smoothness parameter that is notoriously difficult to specify correctly or estimate from the data. In practice, the smoothness parameter is often selected more or less arbitrarily. We introduce rate-unbiasedness, a relaxed notion of asymptotic optimality which requires that the expected ratio of the mean-square error presumed by a potentially misspecified model and the true, but unknown, mean-square error remain bounded away from zero and infinity as more data are obtained. A rate-unbiased model provides uncertainty quantification that is of correct order of magnitude. We then prove that scale estimation suffices for rate-unbiasedness in a variety of common settings. As estimation of the scale of a Gaussian process is routine and requires no optimisation, rate-unbiasedness can be achieved in many applications. |
| title | Scale estimation and rate-unbiasedness for Gaussian processes under smoothness misspecification |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2110.02810 |