Machine learning techniques have received much attention in many areas for regression and classification tasks. In this paper, two new support vector regression (SVR) models, namely, least-square SVR and ε-SVR, are developed under the Bayesian inference framework with a square loss function and a ε-insensitive squared one respectively. In this framework, a Gaussian process prior is assigned to the regression function, and maximum posterior estimate of this function results in a support vector regression problem. The proposed method provides point-wise probabilistic prediction while keeps the structural risk minimization principle, and it allows us to determine the optimal hyper-parameters by maximizing Bayesian model evidence. Based on the Bayesian SVR model, an active learning algorithm is developed, and new training points are selected adaptively based on a learning function to update the SVR model progressively. Numerical results reveal that the developed two Bayesian SVR models are very promising for constructing accurate regression model for problems with diverse characteristics, especially for medium and high dimensional problems.

机器学习技术已在许多领域吸引了很多回归和分类任务的注意。在本文中，两个新的支持向量回归（SVR）模型，即最小二乘SVR和ε- SVR，是在贝叶斯推理框架下开发出一种带有平方损失和ε-不敏感的平方分别为1。在此框架中，将高斯过程先验分配给回归函数，并且该函数的最大后验估计会导致支持向量回归问题。所提出的方法在保持结构风险最小化原则的同时，提供了逐点概率预测，并允许我们通过最大化贝叶斯模型证据来确定最佳超参数。基于贝叶斯SVR模型，开发了一种主动学习算法，并根据学习功能自适应选择新的训练点，以逐步更新SVR模型。数值结果表明，所开发的两个贝叶斯SVR模型对于构建具有多种特征的问题，尤其是中，高维问题的精确回归模型非常有希望。