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Minwoo Chae

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math.ST Statistics Theory Machine Learning

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preprint2026arXiv

The Bernstein-von Mises theorem for Bayesian one-pass online learning

Bayesian online learning provides a coherent framework for sequential inference. However, its theoretical understanding remains limited, particularly in the one-pass setting. Existing theoretical guarantees typically require the mini-batch sample size to diverge, a condition that fails in the one-pass regime. In this paper, we propose a new Bayesian online learning algorithm tailored to the one-pass setting, which incorporates a warm-start phase to ensure stable sequential updates. For this algorithm, we show that the sequentially updated posterior attains the optimal convergence rate. Building on this, we establish an online analogue of the Bernstein-von Mises theorem, which guarantees valid uncertainty quantification without diverging mini-batch sample sizes. Our analysis is based on a novel theoretical framework that differs fundamentally from existing approaches in the online learning literature. Numerical experiments on generalized linear models show that the proposed method matches the performance of the batch estimator while outperforming existing online procedures.

preprint2020arXiv

Bayesian High-dimensional Semi-parametric Inference beyond sub-Gaussian Errors

We consider a sparse linear regression model with unknown symmetric error under the high-dimensional setting. The true error distribution is assumed to belong to the locally $β$-Hölder class with an exponentially decreasing tail, which does not need to be sub-Gaussian. We obtain posterior convergence rates of the regression coefficient and the error density, which are nearly optimal and adaptive to the unknown sparsity level. Furthermore, we derive the semi-parametric Bernstein-von Mises (BvM) theorem to characterize asymptotic shape of the marginal posterior for regression coefficients. Under the sub-Gaussianity assumption on the true score function, strong model selection consistency for regression coefficients are also obtained, which eventually asserts the frequentist's validity of credible sets.

preprint2015arXiv

The semiparametric Bernstein-von Mises theorem for models with symmetric error

In a smooth semiparametric model, the marginal posterior distribution of the finite dimensional parameter of interest is expected to be asymptotically equivalent to the sampling distribution of frequentist's efficient estimators. This is the assertion of the so-called Bernstein-von Mises theorem, and recently, it has been proved in many interesting semiparametric models. In this thesis, we consider the semiparametric Bernstein-von Mises theorem in some models which have symmetric errors. The simplest example of these models is the symmetric location model that has 1-dimensional location parameter and unknown symmetric error. Also, the linear regression and random effects models are included provided the error distribution is symmetric. The condition required for nonparametric priors on the error distribution is very mild, and the most well-known Dirichlet process mixture of normals works well. As a consequence, Bayes estimators in these models satisfy frequentist criteria of optimality such as Hajek-Le Cam convolution theorem. The proof of the main result requires that the expected log likelihood ratio has a certain quadratic expansion, which is a special property of symmetric densities. One of the main contribution of this thesis is to provide an efficient estimator of regression coefficients in the random effects model, in which it is unknown to estimate the coefficients efficiently because the full likelihood inference is difficult. Our theorems imply that the posterior mean or median is efficient, and the result from numerical studies also shows the superiority of Bayes estimators. For practical use of our main results, efficient Gibbs sampler algorithms based on symmetrized Dirichlet process mixtures are provided.