Comparison of sampling techniques for Bayesian parameter estimation

The posterior probability distribution for a set of model parameters encodes all that the data have to tell us in the context of a given model; it is the fundamental quantity for Bayesian parameter estimation. In order to infer the posterior probability distri-bution we have to decide how to explore parameter space. Here we compare three prescriptions for how parameter space is navigated, discussing their relative mer-its. We consider Metropolis-Hasting sampling, nested sampling and affine-invariant ensemble MCMC sampling. We focus on their performance on toy-model Gaussian likelihoods and on a real-world cosmological data set. We outline the sampling al-gorithms themselves and elaborate on performance diagnostics such as convergence time, scope for parallelisation, dimensional scaling, requisite tunings and suitability for non-Gaussian distributions. We find that nested sampling delivers high-fidelity es-timates for posterior statistics at low computational cost, and should be adopted in favour of Metropolis-Hastings in many cases. Affine-invariant MCMC is competitive when computing clusters can be utilised for massive parallelisation. Affine-invariant MCMC and existing extensions to nested sampling naturally probe multi-modal and curving distributions.