  We formulate both Markov chain Monte Carlo (MCMC) sampling algorithms and
basic statistical physics in terms of elementary symmetries. This perspective
on sampling yields derivations of well-known MCMC algorithms and a new parallel
algorithm that appears to converge more quickly than current state of the art
methods. The symmetry perspective also yields a parsimonious framework for
statistical physics and a practical approach to constructing meaningful notions
of effective temperature and energy directly from time series data. We apply
these latter ideas to Anosov systems.
