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Although Bayesian inference is an immensely popular paradigm among a large segment of scientists including statisticians, most applications consider objective priors and need critical investigations (Efron, 2013, Science). While it has…
We study the posterior distribution of the Bayesian multiple change-point regression problem when the number and the locations of the change-points are unknown. While it is relatively easy to apply the general theory to obtain the…
We investigate the frequentist posterior contraction rate of nonparametric Bayesian procedures in linear inverse problems in both the mildly and severely ill-posed cases. A theorem is proved in a general Hilbert space setting under…
This paper analyzes hierarchical Bayesian inverse problems using techniques from high-dimensional statistics. Our analysis leverages a property of hierarchical Bayesian regularizers that we call approximate decomposability to obtain…
In this paper, we study the Tikhonov regularization scheme in Hilbert scales for the nonlinear statistical inverse problem with a general noise. The regularizing norm in this scheme is stronger than the norm in Hilbert space. We focus on…
The choice of tuning parameters in Bayesian variable selection is a critical problem in modern statistics. In particular, for Bayesian linear regression with non-local priors, the scale parameter in the non-local prior density is an…
In this paper, we extend the correspondence between Bayes' estimation and optimal interpolation in a Reproducing Kernel Hilbert Space (RKHS) to the case of linear inequality constraints such as boundedness, monotonicity or convexity. In the…
We investigate the asymptotic behavior of posterior distributions of regression coefficients in high-dimensional linear models as the number of dimensions grows with the number of observations. We show that the posterior distribution…
We demonstrate an equivalence between reproducing kernel Hilbert space (RKHS) embeddings of conditional distributions and vector-valued regressors. This connection introduces a natural regularized loss function which the RKHS embeddings…
Estimation of parameters that obey specific constraints is crucial in statistics and machine learning; for example, when parameters are required to satisfy boundedness, monotonicity, or linear inequalities. Traditional approaches impose…
Variable selection is central to high-dimensional data analysis, and various algorithms have been developed. Ideally, a variable selection algorithm shall be flexible, scalable, and with theoretical guarantee, yet most existing algorithms…
We construct optimal low-rank approximations for the Gaussian posterior distribution in linear Gaussian inverse problems with possibly infinite-dimensional separable Hilbert parameter spaces and finite-dimensional data spaces. We first…
In this paper, we consider the nonparametric least square regression in a Reproducing Kernel Hilbert Space (RKHS). We propose a new randomized algorithm that has optimal generalization error bounds with respect to the square loss, closing a…
Conditional stability estimates require additional regularization for obtaining stable approximate solutions if the validity area of such estimates is not completely known. In this context, we consider ill-posed nonlinear inverse problems…
This paper presents an efficient Bayesian framework for solving nonlinear, high-dimensional model calibration problems. It is based on a Variational Bayesian formulation that aims at approximating the exact posterior by means of solving an…
We generalize Jan Willems' behavioral approach to a class of discrete-time nonlinear systems in a vector-valued reproducing kernel Hilbert space (RKHS). Apart from linear time-invariant systems, this class covers nonlinear systems modeled…
Motivated by applications to the study of stochastic processes, we introduce a new analysis of positive definite kernels $K$, their reproducing kernel Hilbert spaces (RKHS), and an associated family of feature spaces that may be chosen in…
This work presents a nonparametric framework for dissipativity learning in reproducing kernel Hilbert spaces, which enables data-driven certification of stability and performance properties for unknown nonlinear systems without requiring an…
Learning in the reproducing kernel Hilbert space (RKHS) such as the support vector machine has been recognized as a promising technique. It continues to be highly effective and competitive in numerous prediction tasks, particularly in…
Kernel-based machine learning algorithms are based on mapping data from the original input feature space to a kernel feature space of higher dimensionality to solve a linear problem in that space. Over the last decade, kernel based…