Comparing Scale Parameter Estimators for Gaussian Process Interpolation with the Brownian Motion Prior: Leave-One-Out Cross Validation and Maximum Likelihood
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866908497178787840 |
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| author | Naslidnyk, Masha Kanagawa, Motonobu Karvonen, Toni Mahsereci, Maren |
| author_facet | Naslidnyk, Masha Kanagawa, Motonobu Karvonen, Toni Mahsereci, Maren |
| contents | Gaussian process (GP) regression is a Bayesian nonparametric method for regression and interpolation, offering a principled way of quantifying the uncertainties of predicted function values. For the quantified uncertainties to be well-calibrated, however, the kernel of the GP prior has to be carefully selected. In this paper, we theoretically compare two methods for choosing the kernel in GP regression: cross-validation and maximum likelihood estimation. Focusing on the scale-parameter estimation of a Brownian motion kernel in the noiseless setting, we prove that cross-validation can yield asymptotically well-calibrated credible intervals for a broader class of ground-truth functions than maximum likelihood estimation, suggesting an advantage of the former over the latter. Finally, motivated by the findings, we propose interior cross validation, a procedure that adapts to an even broader class of ground-truth functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_07466 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Comparing Scale Parameter Estimators for Gaussian Process Interpolation with the Brownian Motion Prior: Leave-One-Out Cross Validation and Maximum Likelihood Naslidnyk, Masha Kanagawa, Motonobu Karvonen, Toni Mahsereci, Maren Statistics Theory 60G15, 62G20, 68T37, 65D05 Gaussian process (GP) regression is a Bayesian nonparametric method for regression and interpolation, offering a principled way of quantifying the uncertainties of predicted function values. For the quantified uncertainties to be well-calibrated, however, the kernel of the GP prior has to be carefully selected. In this paper, we theoretically compare two methods for choosing the kernel in GP regression: cross-validation and maximum likelihood estimation. Focusing on the scale-parameter estimation of a Brownian motion kernel in the noiseless setting, we prove that cross-validation can yield asymptotically well-calibrated credible intervals for a broader class of ground-truth functions than maximum likelihood estimation, suggesting an advantage of the former over the latter. Finally, motivated by the findings, we propose interior cross validation, a procedure that adapts to an even broader class of ground-truth functions. |
| title | Comparing Scale Parameter Estimators for Gaussian Process Interpolation with the Brownian Motion Prior: Leave-One-Out Cross Validation and Maximum Likelihood |
| topic | Statistics Theory 60G15, 62G20, 68T37, 65D05 |
| url | https://arxiv.org/abs/2307.07466 |