arXiv subjects

Minwoo Chae

Publications and source records attributed to Minwoo Chae.

25 records · Page 2Linked to original sources

arXiv2020

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.

math.ST↗

arXiv2019

Bayesian Sparse Linear Regression with Unknown Symmetric Error

We study full Bayesian procedures for sparse linear regression when errors have a symmetric but otherwise unknown distribution. The unknown error distribution is endowed with a symmetrized Dirichlet process mixture of Gaussians. For the prior on regression coefficients, a mixture of point masses at zero and continuous distributions is considered. We study behavior of the posterior with diverging number of predictors. Conditions are provided for consistency in the mean Hellinger distance. The compatibility and restricted eigenvalue conditions yield the minimax convergence rate of the regression coefficients in $\ell_1$- and $\ell_2$-norms, respectively. The convergence rate is adaptive to both the unknown sparsity level and the unknown symmetric error density under compatibility conditions. In addition, strong model selection consistency and a semi-parametric Bernstein-von Mises theorem are proven under slightly stronger conditions.

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arXiv2018

An EM based Iterative Method for Solving Large Sparse Linear Systems

We propose a novel iterative algorithm for solving a large sparse linear system. The method is based on the EM algorithm. If the system has a unique solution, the algorithm guarantees convergence with a geometric rate. Otherwise, convergence to a minimal Kullback--Leibler divergence point is guaranteed. The algorithm is easy to code and competitive with other iterative algorithms.

math.NA↗

arXiv2017

Fast nonparametric near-maximum likelihood estimation of a mixing density

Mixture models are regularly used in density estimation applications, but the problem of estimating the mixing distribution remains a challenge. Nonparametric maximum likelihood produce estimates of the mixing distribution that are discrete, and these may be hard to interpret when the true mixing distribution is believed to have a smooth density. In this paper, we investigate an algorithm that produces a sequence of smooth estimates that has been conjectured to converge to the nonparametric maximum likelihood estimator. Here we give a rigorous proof of this conjecture, and propose a new data-driven stopping rule that produces smooth near-maximum likelihood estimates of the mixing density, and simulations demonstrate the quality empirical performance of this estimator.

stat.CO↗

arXiv2017

Applications of an algorithm for solving Fredholm equations of the first kind

In this paper we use an iterative algorithm for solving Fredholm equations of the first kind. The basic algorithm is known and is based on an EM algorithm when involved functions are non-negative and integrable. With this algorithm we demonstrate two examples involving the estimation of a mixing density and a first passage time density function involving Brownian motion. We also develop the basic algorithm to include functions which are not necessarily non-negative and again present illustrations under this scenario. A self contained proof of convergence of all the algorithms employed is presented.

math.ST↗

arXiv2017

The semi-parametric Bernstein-von Mises theorem for regression models with symmetric errors

In a smooth semi-parametric model, the marginal posterior distribution for a finite dimensional parameter of interest is expected to be asymptotically equivalent to the sampling distribution of any efficient point-estimator. The assertion leads to asymptotic equivalence of credible and confidence sets for the parameter of interest and is known as the semi-parametric Bernstein-von Mises theorem. In recent years, it has received much attention and has been applied in many examples. We consider models in which errors with symmetric densities play a role; more specifically, it is shown that the marginal posterior distributions of regression coefficients in the linear regression and linear mixed effect models satisfy the semi-parametric Bernstein-von Mises assertion. As a consequence, Bayes estimators in these models achieve frequentist inferential optimality, as expressed e.g. through Hajek's convolution and asymptotic minimax theorems. Conditions for the prior on the space of error densities are relatively mild and well-known constructions like the Dirichlet process mixture of normal densities and random series priors constitute valid choices. Particularly, the result provides an efficient estimate of regression coefficients in the linear mixed effect model, for which no other efficient point-estimator was known previously.

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arXiv2015

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.

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