In Bayesian inference,the computation of high-dimensional integral for posterior distributions is a conundrum.A dominant paradigm for solving this problem is to use stochastic methods like MCMC.However,MCMC suffers from low computational efficiency for large high-dimensional data sets or complex models.Also,it is hard to determine its convergence.This paper proposes a new efficient variational Bayes(VB)algorithm for parameter estimation and variable selection in Bayesian adaptive shrinkage,Bayesian LASSO,and extended Bayesian LASSO.Originated from the mean-field theory in theoretical physics,VB algorithm approximates the target posterior distribution by using the closest distribution in a specified distribution family with an easy-to-assess convergence criterion.Simulations suggest that the proposed VB algorithm exhibits competitive performance in estimation accuracy and variable selection with higher speed compared with those of MCMC algorithms.We demonstrate VB in the analysis of the price of the real estate market in Russia.
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