贝叶斯场论：密度估计、回归、分类和逆量子问题的非参数方法
Bayesian Field Theory: Nonparametric Approaches to Density Estimation, Regression, Classification, and Inverse Quantum Problems
摘要：
贝叶斯场论表示从观测数据学习函数的非参数贝叶斯方法。基于贝叶斯统计的原理，通过组合两个模型来定义特定的贝叶斯场论：似然模型，提供测量过程的概率描述，以及先验模型，提供从训练数据到非训练数据的概括所需的信息。本文中讨论的特定似然模型是一般密度估计、高斯回归、聚类、分类和特定于逆量子问题的模型。除了问题典型的硬约束，如概率的归一化和正性，先验模型必须实现所有特定的，往往是模糊的，“先验”知识可用于特定的任务。本文讨论的非参数先验模型是高斯过程、高斯混合过程和非二次势。先前的模型通过包括超参数而变得灵活。特别是，高斯过程分量的均值函数和协方差算子的自适应进行了详细讨论。即使使用高斯过程构建块来构建，贝叶斯场论也通常是非高斯的，因此必须数值求解。随着计算资源的增加，具有实际意义的、数值上可行的非高斯贝叶斯场论的种类也在稳步增长。那些在计算上要求太高的模型可以作为出发点来构建更容易解决的参数方法，例如使用变分技术。
英文摘要：
Bayesian field theory denotes a nonparametric Bayesian approach for learning functions from observational data. Based on the principles of Bayesian statistics, a particular Bayesian field theory is defined by combining two models: a likelihood model, providing a probabilistic description of the measurement process, and a prior model, providing the information necessary to generalize from training to non-training data. The particular likelihood models discussed in the paper are those of general density estimation, Gaussian regression, clustering, classification, and models specific for inverse quantum problems. Besides problem typical hard constraints, like normalization and positivity for probabilities, prior models have to implement all the specific, and often vague, "a priori" knowledge available for a specific task. Nonparametric prior models discussed in the paper are Gaussian processes, mixtures of Gaussian processes, and non-quadratic potentials. Prior models are made flexible by including hyperparameters. In particular, the adaption of mean functions and covariance operators of Gaussian process components is discussed in detail. Even if constructed using Gaussian process building blocks, Bayesian field theories are typically non-Gaussian and have thus to be solved numerically. According to increasing computational resources the class of non-Gaussian Bayesian field theories of practical interest which are numerically feasible is steadily growing. Models which turn out to be computationally too demanding can serve as starting point to construct easier to solve parametric approaches, using for example variational techniques.
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