  This work develops a powerful and versatile framework for determining
acceptance ratios in Metropolis-Hastings type Markov kernels widely used in
statistical sampling problems. Our approach allows us to derive new classes of
kernels which unify random walk or diffusion-type sampling methods with more
complicated "extended phase space" algorithms based around ideas from
Hamiltonian dynamics. Our starting point is an abstract result developed in the
generality of measurable state spaces that addresses proposal kernels that
possess a certain involution structure. Note that, while this underlying
proposal structure suggests a scope which includes Hamiltonian-type kernels, we
demonstrate that our abstract result is, in an appropriate sense, equivalent to
an earlier general state space setting developed in [Tierney, Annals of Applied
Probability, 1998] where the connection to Hamiltonian methods was more
obscure. Altogether, the theoretical unity and reach of our main result
provides a basis for deriving novel sampling algorithms while laying bare
important relationships between existing methods.
