In many fields of social and industrial sciences, simulation is crucial in comprehending a target system. A major task in simulation is the estimation of optimal parameters to express the observed data need to directly elucidate the properties of the target system as a modeling based on the expert's domain knowledge. However, skilled human experts struggle to obtain the desired parameters. Data assimilation therefore becomes an unavoidable task to reduce the cost of simulator optimization. Another necessary task is extrapolation; in many practical cases, predictions based on simulation results will be often outside of the dominant range of a given data area, and this is referred to as the covariate shift. This paper focuses on a regression problem with covariate shift. While the parameter estimation for the covariate shift has been studied thoroughly in parametric and nonparametric settings, conventional statistical methods of parameter searching are not applicable in the data assimilation of the simulation owing to the properties of the likelihood function: intractable or nondifferentiable. Hence, we propose a novel framework of Bayesian inference based on kernel mean embedding. This framework allows for predictions in covariate shift situations, and its effectiveness is evaluated in both synthetic numerical experiments and a widely used production simulator reproducing real-world manufacturing factories.
Kernel mean embedding is a framework to map distributions into a reproducing kernel Hilbert space (RKHS) @math as a feature space @cite_7 . Figure shows a schematic illustration of the kernel mean embedding. In this section, we briefly review three applications of kernel mean embedding: kernel ABC, kernel sum rule, and kernel herding. The detailed formulations of these methods are described in with the proposed method.
@cite_7
2418306335 2044514896 2124331852 1986007546
A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference. The basic idea behind this framework is to map distributions into a reproducing kernel Hilbert space (RKHS) in which the whole arsenal of kernel methods can be extended to probability measures. It can be viewed as a generalization of the original "feature map" common to support vector machines (SVMs) and other kernel methods. While initially closely associated with the latter, it has meanwhile found application in fields ranging from kernel machines and probabilistic modeling to statistical inference, causal discovery, and deep learning. The goal of this survey is to give a comprehensive review of existing work and recent advances in this research area, and to discuss the most challenging issues and open problems that could lead to new research directions. The survey begins with a brief introduction to the RKHS and positive definite kernels which forms the backbone of this survey, followed by a thorough discussion of the Hilbert space embedding of marginal distributions, theoretical guarantees, and a review of its applications. The embedding of distributions enables us to apply RKHS methods to probability measures which prompts a wide range of applications such as kernel two-sample testing, independent testing, and learning on distributional data. Next, we discuss the Hilbert space embedding for conditional distributions, give theoretical insights, and review some applications. The conditional mean embedding enables us to perform sum, product, and Bayes' rules---which are ubiquitous in graphical model, probabilistic inference, and reinforcement learning---in a non-parametric way. We then discuss relationships between this framework and other related areas. Lastly, we give some suggestions on future research directions. Kernel methods can embed finite-dimensional data into infinite-dimensional feature spaces. In spite of the large underlying feature dimensionality, kernel methods can achieve good generalization ability. This observation is often wrongly interpreted, and it has been used to argue that kernel learning can magically avoid the "curse-of-dimensionality" phenomenon encountered in statistical estimation problems. This letter shows that although using kernel representation, one can embed data into an infinite-dimensional feature space; the effective dimensionality of this embedding, which determines the learning complexity of the underlying kernel machine, is usually small. In particular, we introduce an algebraic definition of a scale-sensitive effective dimension associated with a kernel representation. Based on this quantity, we derive upper bounds on the generalization performance of some kernel regression methods. Moreover, we show that the resulting convergent rates are optimal under various circumstances. A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing, and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). A pseudometric on the space of probability measures can be defined as the distance between distribution embeddings: we denote this as γk, indexed by the kernel function k that defines the inner product in the RKHS. We present three theoretical properties of γk. First, we consider the question of determining the conditions on the kernel k for which γk is a metric: such k are denoted characteristic kernels. Unlike pseudometrics, a metric is zero only when two distributions coincide, thus ensuring the RKHS embedding maps all distributions uniquely (i.e., the embedding is injective). While previously published conditions may apply only in restricted circumstances (e.g., on compact domains), and are difficult to check, our conditions are straightforward and intuitive: integrally strictly positive definite kernels are characteristic. Alternatively, if a bounded continuous kernel is translation-invariant on ℜd, then it is characteristic if and only if the support of its Fourier transform is the entire ℜd. Second, we show that the distance between distributions under γk results from an interplay between the properties of the kernel and the distributions, by demonstrating that distributions are close in the embedding space when their differences occur at higher frequencies. Third, to understand the nature of the topology induced by γk, we relate γk to other popular metrics on probability measures, and present conditions on the kernel k under which γk metrizes the weak topology. Kernel k-means and spectral clustering have both been used to identify clusters that are non-linearly separable in input space. Despite significant research, these methods have remained only loosely related. In this paper, we give an explicit theoretical connection between them. We show the generality of the weighted kernel k-means objective function, and derive the spectral clustering objective of normalized cut as a special case. Given a positive definite similarity matrix, our results lead to a novel weighted kernel k-means algorithm that monotonically decreases the normalized cut. This has important implications: a) eigenvector-based algorithms, which can be computationally prohibitive, are not essential for minimizing normalized cuts, b) various techniques, such as local search and acceleration schemes, may be used to improve the quality as well as speed of kernel k-means. Finally, we present results on several interesting data sets, including diametrical clustering of large gene-expression matrices and a handwriting recognition data set.