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.

翻译：在输入-输出关系的概率监督学习中——作为高斯过程样本函数——通常为核的超参数指定先验，该核参数化高斯过程的协方差函数，而由此产生的（多元正态）似然的诱导协方差矩阵控制着学习和预测。当所求函数高度多元时，必须同时学习多个长度尺度参数，这使得推理变得困难。我们为协方差矩阵开发了一种"自组装"Wishart先验，同时使用MCMC对核超参数进行贝叶斯推断。该构造利用最近MCMC迭代的回顾窗口来定义依赖于时间步长的尺度矩阵，从而为链引入自适应性。结果表明，直接对协方差矩阵指定先验有助于在高斯过程学习范式中诊断弱信息输入。我们通过两个不同的实证例子支持我们的先验开发——一个基于合成数据，另一个基于真实世界数据集。