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Jaeyong Lee

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Methodology Machine Learning math.ST Applications Statistics Theory

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preprint2026arXiv

Posterior Contraction of Lévy Adaptive B-spline Regression in Besov Spaces

We investigate the asymptotic properties of the Lévy Adaptive B-spline (LABS) regression model, a Bayesian nonparametric method that incorporates B-spline kernels into the Lévy Adaptive Regression Kernel (LARK) model. LABS applies splines of varying degrees with independently defined knots, yielding a flexible model class capable of adapting to irregular and locally structured features of the true function. Within the nonparametric regression framework with univariate random design and Gaussian errors, we establish that the LABS posterior contracts around the true function in Besov classes at nearly minimax-optimal rates, up to a logarithmic factor, while adapting automatically to unknown smoothness. This study contributes to filling a gap in the literature, where theoretical results on posterior contraction of the LARK model in Besov spaces remain scarce. Simulation experiments on standard test functions in Besov spaces, including Blocks, Bumps, HeaviSine, and Doppler, complement the theoretical results and demonstrate the practical utility of LABS.

preprint2026arXiv

Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces

We study posterior contraction rates for sparse Bayesian Kolmogorov-Arnold networks (KANs) over anisotropic Besov spaces, providing a statistical foundation of KANs from a Bayesian point of view. We show that sparse Bayesian KANs equipped with spike-and-slab-type sparsity priors attain the near-minimax posterior contraction. In particular, the contraction rate depends on the intrinsic anisotropic smoothness of the underlying function. Moreover, by placing a hyperprior on a single model-size parameter, the resulting posterior adapts to unknown anisotropic smoothness and still achieves the corresponding near-minimax rate. A distinctive feature of our results, compared with those for standard sparse MLP-based models, is that the KAN depth can be kept fixed: owing to the flexibility of learnable spline edge functions, the required approximation complexity is controlled through the network width, spline-grid range and size, and parameter sparsity. Our analysis develops theoretical tools tailored to sparse spline-edge architectures, including approximation and complexity bounds for Bayesian KANs. We then extend to compositional Besov spaces and show that the contraction rates depend on layerwise smoothness and effective dimension of the underlying compositional structure, thereby effectively avoiding the curse of dimensionality. Together, the developed tools and findings advance the theoretical understanding of Bayesian neural networks and provide rigorous statistical foundations for KANs.

preprint2022arXiv

Estimation of World Seroprevalence of SARS-CoV-2 antibodies

In this paper, we estimate the seroprevalence against COVID-19 by country and derive the seroprevalence over the world. To estimate seroprevalence, we use serological surveys (also called the serosurveys) conducted within each country. When the serosurveys are incorporated to estimate world seroprevalence, there are two issues. First, there are countries in which a serological survey has not been conducted. Second, the sample collection dates differ from country to country. We attempt to tackle these problems using the vaccination data, confirmed cases data, and national statistics. We construct Bayesian models to estimate the numbers of people who have antibodies produced by infection or vaccination separately. For the number of people with antibodies due to infection, we develop a hierarchical model for combining the information included in both confirmed cases data and national statistics. At the same time, we propose regression models to estimate missing values in the vaccination data. As of 31st of July 2021, using the proposed methods, we obtain the 95% credible interval of the world seroprevalence as [38.6%, 59.2%].

preprint2021arXiv

Lévy Adaptive B-spline Regression via Overcomplete Systems

The estimation of functions with varying degrees of smoothness is a challenging problem in the nonparametric function estimation. In this paper, we propose the LABS (Lévy Adaptive B-Spline regression) model, an extension of the LARK models, for the estimation of functions with varying degrees of smoothness. LABS model is a LARK with B-spline bases as generating kernels. The B-spline basis consists of piecewise k degree polynomials with k-1 continuous derivatives and can express systematically functions with varying degrees of smoothness. By changing the orders of the B-spline basis, LABS can systematically adapt the smoothness of functions, i.e., jump discontinuities, sharp peaks, etc. Results of simulation studies and real data examples support that this model catches not only smooth areas but also jumps and sharp peaks of functions. The proposed model also has the best performance in almost all examples. Finally, we provide theoretical results that the mean function for the LABS model belongs to the certain Besov spaces based on the orders of the B-spline basis and that the prior of the model has the full support on the Besov spaces.

preprint2021arXiv

Seroprevalence of SARS-CoV-2 antibodies in South Korea

In $2020$, Korea Disease Control and Prevention Agency reported three rounds of surveys on seroprevalence of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) antibodies in South Korea. We analyze the seroprevalence surveys using a Bayesian method with an informative prior distribution on the seroprevalence parameter, and the sensitivity and specificity of the diagnostic test. We construct the informative prior using the posterior distribution obtained from the clinical evaluation data based on the plaque reduction neutralization test. The constraint of the seroprevalence parameter induced from the known confirmed coronavirus 2019 cases can be imposed naturally in the proposed Bayesian model. We also prove that the confidence interval of the seroprevalence parameter based on the Rao's test can be the empty set, while the Bayesian method renders a reasonable interval estimator. As of the $30$th of October $2020$, the $95\%$ credible interval of the estimated SARS-CoV-2 positive population does not exceed $307,448$, approximately $0.6\%$ of the Korean population.

preprint2021arXiv

The Beta-Mixture Shrinkage Prior for Sparse Covariances with Posterior Minimax Rates

Statistical inference for sparse covariance matrices is crucial to reveal dependence structure of large multivariate data sets, but lacks scalable and theoretically supported Bayesian methods. In this paper, we propose beta-mixture shrinkage prior, computationally more efficient than the spike and slab prior, for sparse covariance matrices and establish its minimax optimality in high-dimensional settings. The proposed prior consists of beta-mixture shrinkage and gamma priors for off-diagonal and diagonal entries, respectively. To ensure positive definiteness of the resulting covariance matrix, we further restrict the support of the prior to a subspace of positive definite matrices. We obtain the posterior convergence rate of the induced posterior under the Frobenius norm and establish a minimax lower bound for sparse covariance matrices. The class of sparse covariance matrices for the minimax lower bound considered in this paper is controlled by the number of nonzero off-diagonal elements and has more intuitive appeal than those appeared in the literature. The obtained posterior convergence rate coincides with the minimax lower bound unless the true covariance matrix is extremely sparse. In the simulation study, we show that the proposed method is computationally more efficient than competitors, while achieving comparable performance. Advantages of the shrinkage prior are demonstrated based on two real data sets.

preprint2016arXiv

Laplace based approximate posterior inference for differential equation models

Ordinary differential equations are arguably the most popular and useful mathematical tool for describing physical and biological processes in the real world. Often, these physical and biological processes are observed with errors, in which case the most natural way to model such data is via regression where the mean function is defined by an ordinary differential equation believed to provide an understanding of the underlying process. These regression based dynamical models are called differential equation models. Parameter inference from differential equation models poses computational challenges mainly due to the fact that analytic solutions to most differential equations are not available. In this paper, we propose an approximation method for obtaining the posterior distribution of parameters in differential equation models. The approximation is done in two steps. In the first step, the solution of a differential equation is approximated by the general one-step method which is a class of numerical methods for ordinary differential equations including the Euler and the Runge-Kutta procedures; in the second step, nuisance parameters are marginalized using Laplace approximation. The proposed Laplace approximated posterior gives a computationally fast alternative to the full Bayesian computational scheme (such as Markov Chain Monte Carlo) and produces more accurate and stable estimators than the popular smoothing methods (called collocation methods) based on frequentist procedures. For a theoretical support of the proposed method, we prove that the Laplace approximated posterior converges to the actual posterior under certain conditions and analyze the relation between the order of numerical error and its Laplace approximation. The proposed method is tested on simulated data sets and compared with the other existing methods.

preprint2013arXiv

Defining Predictive Probability Functions for Species Sampling Models

We review the class of species sampling models (SSM). In particular, we investigate the relation between the exchangeable partition probability function (EPPF) and the predictive probability function (PPF). It is straightforward to define a PPF from an EPPF, but the converse is not necessarily true. In this paper we introduce the notion of putative PPFs and show novel conditions for a putative PPF to define an EPPF. We show that all possible PPFs in a certain class have to define (unnormalized) probabilities for cluster membership that are linear in cluster size. We give a new necessary and sufficient condition for arbitrary putative PPFs to define an EPPF. Finally, we show posterior inference for a large class of SSMs with a PPF that is not linear in cluster size and discuss a numerical method to derive its PPF.

preprint2013arXiv

Generalized double Pareto shrinkage

We propose a generalized double Pareto prior for Bayesian shrinkage estimation and inferences in linear models. The prior can be obtained via a scale mixture of Laplace or normal distributions, forming a bridge between the Laplace and Normal-Jeffreys' priors. While it has a spike at zero like the Laplace density, it also has a Student's $t$-like tail behavior. Bayesian computation is straightforward via a simple Gibbs sampling algorithm. We investigate the properties of the maximum a posteriori estimator, as sparse estimation plays an important role in many problems, reveal connections with some well-established regularization procedures, and show some asymptotic results. The performance of the prior is tested through simulations and an application.

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