Say we are trying to estimate the posterior over a set of random variables using MCMC for a Bayesian model. We have prior knowledge about the variables and we can express this knowledge as a prior pdf.
Now, say this pdf admits (non-trivially) multiple parameterizations (i.e. different mathematical formulas that render the same exact pdf). Do MCMC solvers exploit in practice the parameterization chosen for the prior in any way? And would those differences respond to theoretical or technical reasons?