  Standard Markov chain Monte Carlo (MCMC) admits three fundamental control
parameters: the number of chains, the length of the warmup phase, and the
length of the sampling phase. These control parameters play a large role in
determining the amount of computation we deploy. In practice, we need to walk a
line between achieving sufficient precision and not wasting precious
computational resources and time. We review general strategies to check the
length of the warmup and sampling phases, and examine the three control
parameters of MCMC in the contexts of CPU- and GPU-based hardware. Our
discussion centers around three tasks: (1) inference about a latent variable,
(2) computation of expectation values and quantiles, and (3) diagnostics to
check the reliability of the estimators.
  This chapter begins with general recommendations on the control parameters of
MCMC, which have been battle-tested over the years and often motivate defaults
in Bayesian statistical software. Usually we do not know ahead of time how a
sampler will interact with a target distribution, and so the choice of MCMC
algorithm and its control parameters, tend to be based on experience,
re-evaluated after simulations have been obtained and analyzed. The second part
of this chapter provides a theoretical motivation for our recommended approach,
with pointers to some concerns and open problems. We also examine recent
developments on the algorithmic and hardware fronts, which motivate new
computational approaches to MCMC.
