Conditions For Posterior Consistency Research Articles

High-dimensional posterior consistency of the Bayesian lasso

ABSTRACTThis paper considers posterior consistency in the context of high-dimensional variable selection using the Bayesian lasso algorithm. In a frequentist setting, consistency is perhaps the most basic property that we expect any reasonable estimator to achieve. However, in a Bayesian setting, consistency is often ignored or taken for granted, especially in more complex hierarchical Bayesian models. In this paper, we have derived sufficient conditions for posterior consistency in the Bayesian lasso model with the orthogonal design, where the number of parameters grows with the sample size.

Posterior Consistency of Bayesian Quantile Regression Under a Mis-Specified Likelihood Based on Asymmetric Laplace Density

We provide a theoretical justification for the widely used and yet only empirically verified approach of using Asymmetric Laplace Density(ALD) in Bayesian Quantile Regression. We derive sufficient conditions for posterior consistency of the quantile regression parameters even if the true underlying likelihood is not ALD, by considering both the case of random as well as non-random covariates. While existing literature on misspecified models address more general models, our approach of exploiting the specific form of ALD allows for a more direct derivation. We verify that the conditions so derived are satisfied by a wide range of potential true underlying probability distributions. We also show that posterior consistency holds even in the case of improper priors as long as the posterior is well defined. We demonstrate the working of the method using simulations.

A Direct Approach to Understanding Posterior Consistency of Bayesian Regression Problems

Previous approaches to establishing posterior consistency of Bayesian regression problems have used general theorems that involve verifying sufficient conditions for posterior consistency. In this article, we consider a direct approach by computing the posterior density explicitly and evaluating its asymptotic behavior. For this purpose, we deal with a sample size dependent prior based on a truncated regression function with increasing sample size, and evaluate the asymptotic properties of the resulting posterior. Based on a concept called posterior density consistency, we attempt to understand posterior consistency. As an application, we illustrate that the posterior density of an orthogonal semiparametric regression model is consistent.