We consider the accuracy of an approximate posterior distribution in nonparametric regression problems by combining posterior distributions computed on subsets of the data defined by the locations of the independent variables. We show that this approximate posterior retains the rate of recovery of the full data posterior distribution, where the rate of recovery adapts to the smoothness of the true regression function. As particular examples we consider Gaussian process priors based on integrated Brownian motion and the Mat\'ern kernel augmented with a prior on the length scale. Besides theoretical guarantees we present a numerical study of the methods both on synthetic and real world data. We also propose a new aggregation technique, which numerically outperforms previous approaches.

翻译：我们研究了非参数回归问题中近似后验分布的精度，通过合并由自变量位置定义的数据子集上计算得到的后验分布。我们证明了该近似后验分布保留了完整数据后验分布的恢复速率，且该恢复速率能够自适应真实回归函数的平滑程度。作为具体实例，我们考虑了基于积分布朗运动和附加长度尺度先验的Matérn核的高斯过程先验。除理论保证外，我们还对合成数据和真实世界数据进行了数值研究。此外，我们提出了一种新的聚合技术，其数值表现优于已有方法。