  Reparameterizing a probabilisitic system is common advice for improving the
performance of a statistical algorithm like Markov chain Monte Carlo, even
though in theory such reparameterizations should leave the system, and the
performance of any algorithm, invariant. In this paper I show how the
reparameterizations common in practice are only incomplete reparameterizations
which result in different interactions between a target probabilistic system
and a given algorithm. I then consider how these changing interactions manifest
in the context of Markov chain Monte Carlo algorithms defined on Riemannian
manifolds. In particular I show how any incomplete reparameterization is
equivalent to modifying the metric geometry directly.
