  The Gibbs sampler of Park and Casella is one of the most popular MCMC methods
for sampling from the posterior density of the Bayesian Lasso regression. As
with many Markov chain samplers, their Gibbs sampler lacks a theoretically
sound method of output analysis --- a method for estimating the variance of a
given ergodic average and estimating how closely the chain is sampling from the
stationary distribution, that is, the burn-in.
  In this paper, we address this shortcoming by identifying regenerative
structure in the sampler of Park and Casella, thus providing a theoretically
sound method of assessing its performance. The regenerative structure provides
both a strongly consistent variance estimator, and an estimator of (an upper
bound on) the total variation distance from the target posterior density. The
result is a simple and theoretically sound way to assess the stationarity of
the Park and Casella and, more generally, other MCMC samplers, for which
regenerative simulation is possible.
  We perform a numerical study in which we validate the standard errors
calculated by our regenerative method by comparing it with the standard errors
calculated by an AR(1) heuristic approximation. Thus, we show that for the
Bayesian Lasso model, the regenerative method is a viable and theoretically
justified alternative to the existing ad-hoc MCMC convergence diagnostics.
