Inference in Bayesian regression is often made using Markov chain Monte Carlo (MCMC) methods. For MCMC to perform well, its rate of convergence to the target distribution must be fast. In particular, geometric rates are of specific interest since they allow for output analysis of the MCMC samples. We discuss some results on the rate of MCMC samplers for different Bayesian penalized regression and variable selection models. We focus on the need for relating the rates of convergence of MCMC samplers to controllable parameters of the model so as to allow practitioners to realistically improve speed of samplers.