为高维输入的高斯过程学习设计自适应先验，提升多尺度参数推断效果
Gaussian Process-based learning with new MCMC-based implementation of Wishart prior on correlation matrix
- 基于马尔可夫链最近迭代构建时变尺度矩阵，实现自适应的威沙特先验
- 在合成与真实数据上验证了该先验对弱信息输入的诊断能力
- 适合需要多尺度建模且数据维度高的贝叶斯学习场景
在基于高斯过程（GP）的监督学习中，通常对核函数的超参数设定先验，其诱导的协方差矩阵决定模型的学习与预测。当目标函数高度多维时，需同时学习多个长度尺度参数，导致推断困难。本文提出一种“自组装”威沙特先验，用于协方差矩阵建模，并结合马尔可夫链蒙特卡洛（MCMC）进行核超参数的贝叶斯推断。该方法利用最近若干次MCMC迭代的历史信息构造时间依赖的尺度矩阵，使先验具有自适应性。实验结果表明，直接在协方差矩阵上设定先验有助于识别高斯过程学习中的弱信息输入。研究通过两个独立实证分析——合成数据与真实数据集——验证了所提先验的有效性。
原文摘要 · Abstract (English)
In probabilstic supervised learning of an input-output relationship - as a sample function of a Gaussian Process (GP) - priors are typically specified for the hyperparameters of the kernel that parametrises the covariance function of the GP, where the induced covariance matrix of the (resulting multivariate Normal) likelihood, governs the learning and prediction. When the sought function is highly multivariate, multiple lengthscale parameters must be learnt simultaneously, making inference difficult. We develop a ``self-assembled'' Wishart prior for the covariance matrix, while undertaking Bayesian inference on the kernel hyperparameters using MCMC. The construction uses a look-back window over recent MCMC iterations to define a time-step dependent scale matrix, thereby introducing adaptiveness to the chain. Results suggest that direct prior specification on the covariance matrix can be useful for diagnosing weakly informative inputs within the GP-based learning paradigm. We support our prior development with two distinct empirical illustrations - one on synthetic data, and another on a real-world dataset.
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