Abstract
We consider three Bayesian penalized regression models and show that the respective deterministic scan Gibbs samplers are geometrically ergodic regardless of the dimension of the regression problem. We prove geometric ergodicity of the Gibbs samplers for the Bayesian fused lasso, the Bayesian group lasso, and the Bayesian sparse group lasso. Geometric ergodicity along with a moment condition results in the existence of a Markov chain central limit theorem for Monte Carlo averages and ensures reliable output analysis. Our results of geometric ergodicity allow us to also provide default starting values for the Gibbs samplers.
Highlights
Let y ∈ Rn be the observed realization of the response Y, X be the n × p model matrix, and β ∈ Rp be the regression coefficient vector
We show that the Markov chain Monte Carlo (MCMC) samplers used in the three models converge to their respective stationary distribution at a geometric rate
We show that all three Gibbs samplers converge to their respective stationary distribution at a geometric rate under reasonable conditions
Summary
Let y ∈ Rn be the observed realization of the response Y , X be the n × p model matrix, and β ∈ Rp be the regression coefficient vector. We show that the MCMC samplers used in the three models converge to their respective stationary distribution at a geometric rate. We show that all three Gibbs samplers converge to their respective stationary distribution at a geometric rate under reasonable conditions. We only require the number of observations, n, to be larger than three and require no assumptions on the number of covariates, p or the model matrix X This geometric rate of convergence allows for reliable estimation of posterior quantities in the following way. If the deterministic scan Gibbs sampler is geometrically ergodic and g(β, η, σ2) 2+δ f (β, η, σ2 | y)dβ dη dσ2 < ∞ , a Markov chain CLT holds as below:. Johnson and Jones (2015) established geometric ergodicity of a four variable random scan Gibbs sampler for a hierarchical random effects model. This will lead us to default starting values for the three Gibbs sampler
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