  From basic considerations of the Lie group that preserves a target
probability measure, we derive the Barker, Metropolis, and ensemble Markov
chain Monte Carlo (MCMC) algorithms, as well as variants of waste-recycling
Metropolis-Hastings and an altogether new MCMC algorithm. We illustrate these
constructions with explicit numerical computations, and we empirically
demonstrate on a spin glass that the new algorithm converges more quickly than
its siblings.
