  Metropolis Hastings nested sampling evolves a Markov chain, accepting new
points along the chain according to a version of the Metropolis Hastings
acceptance ratio, which has been modified to satisfy the nested sampling
likelihood constraint. The geometric nested sampling algorithm I present here
is based on the Metropolis Hastings method, but treats parameters as though
they represent points on certain geometric objects, namely circles, tori and
spheres. For parameters which represent points on a circle or torus, the trial
distribution is "wrapped" around the domain of the posterior distribution such
that samples cannot be rejected automatically when evaluating the Metropolis
ratio due to being outside the sampling domain. Furthermore, this enhances the
mobility of the sampler. For parameters which represent coordinates on the
surface of a sphere, the algorithm transforms the parameters into a Cartesian
coordinate system before sampling which again makes sure no samples are
automatically rejected, and provides a physically intuitive way of the sampling
the parameter space.
