高斯过程在平滑度误设定下的尺度估计与速率无偏性
Scale estimation and rate-unbiasedness for Gaussian processes under smoothness misspecification
AI总结：
针对高斯过程回归中平滑度参数误设定的问题，提出速率无偏性概念，证明尺度估计足以实现该性质，从而无需优化即可获得正确数量级的不确定性量化。
摘要：
高斯过程回归在统计和机器学习中广泛用于预测和不确定性量化。高斯过程由其均值函数和协方差函数指定。许多协方差函数（包括Matérn协方差）具有平滑度参数，该参数以难以正确指定或从数据中估计而著称。在实践中，平滑度参数通常或多或少地任意选择。我们引入了速率无偏性，这是一种渐近最优性的放宽概念，要求潜在误设定模型所假定的均方误差与真实但未知的均方误差的期望比值，在获得更多数据时保持有界且远离零和无穷大。速率无偏模型提供数量级正确的不确定性量化。然后，我们证明在各种常见设置中，尺度估计足以实现速率无偏性。由于高斯过程尺度的估计是常规操作且无需优化，因此在许多应用中可以实现速率无偏性。
英文摘要：
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.
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