A variational Bayes approach to inference for low-dimensional parameters in high-dimensional linear regression
高维线性回归中低维目标参数推断的变分贝叶斯方法AI 解读 ↓
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高维线性回归中低维目标参数推断的变分贝叶斯方法AI 解读 ↓
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We propose a scalable variational Bayes method for statistical inference for a single or pre-specified low-dimensional subset of the coordinates of a high-dimensional parameter in sparse linear regression. Our approach relies on assigning a mean-field approximation to the nuisance coordinates and carefully modelling the conditional distribution of the target given the nuisance. This requires only a preprocessing step and preserves the computational advantages of mean-field variational Bayes, while ensuring accurate and reliable inference for the target parameter, including for uncertainty quantification. We investigate the numerical performance of our algorithm, showing that it performs competitively with existing methods. We further establish accompanying theoretical guarantees for estimation and uncertainty quantification in the form of a Bernstein--von Mises theorem for a possibly growing subset.
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RBF 核高斯过程最大似然估计的精确渐近理论
Gaussian processes (GPs) are widely used across machine learning, spatial statistics, time-series analysis, optimization, Bayesian statistics, and scientific applications. A central component of a GP model is its kernel, which is typically specified through a parametric family. Among the most widely used choices is the radial basis function (RBF), also known as the squared exponential or Gaussian kernel, owing to its simple form, smoothness, and flexibility. In practice, the…
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We study the fundamental limits of convex-regularized estimation in high-dimensional linear regression with right-rotationally invariant design matrices. We show that the asymptotic risks of convex-penalized least squares estimators are lower bounded by the risk of an approximate message passing algorithm known as Bayes VAMP, and we further characterize when the lower bound is attainable. As a technical ingredient in the proof of our lower bound theorem, we characterize the…
基于相容性的贝叶斯推断:渐近极限、变分原理与 Wasserstein 几何
Approximate Bayesian Computation (ABC) replaces intractable likelihood evaluations with comparisons between observed and simulated data. We develop a measure-theoretic formulation of discrepancy-based Bayesian inference in which compatibility is encoded through nonnegative weights. For vanishing thresholds, we establish convergence of the corresponding posterior distributions in total variation. In the large-sample regime, we identify conditions under which the posterior…
大规模需求估计的贝叶斯机器学习方法
This work studies how Bayesian machine learning methods can be used for large-scale demand estimation with many product categories. I compare two model classes, a latent factorization model and a mixed logit model and two Bayesian estimation approaches, Markov Chain Monte Carlo (MCMC) and Variational Inference (VI). The analysis combines a simulation study with an application to supermarket scanner data. The results show that the latent factorization model benefits from…
分数岭回归
Regularization methods such as the Lasso combine shrinkage and variable selection, but strong penalization drives their solutions toward the null model. We introduce Fractional Ridge (Fridge) regression, a penalized regression method that instead shrinks toward models of a prespecified target size. Its penalty vanishes whenever the number of nonzero coefficients does not exceed that target. This behavior connects continuous shrinkage with best subset selection. We develop an…