  The use of Cauchy Markov random field priors in statistical inverse problems
can potentially lead to posterior distributions which are non-Gaussian,
high-dimensional, multimodal and heavy-tailed. In order to use such priors
successfully, sophisticated optimization and Markov chain Monte Carlo (MCMC)
methods are usually required. In this paper, our focus is largely on reviewing
recently developed Cauchy difference priors, while introducing interesting new
variants, whilst providing a comparison. We firstly propose a one-dimensional
second order Cauchy difference prior, and construct new first and second order
two-dimensional isotropic Cauchy difference priors. Another new Cauchy prior is
based on the stochastic partial differential equation approach, derived from
Mat\'{e}rn type Gaussian presentation. The comparison also includes Cauchy
sheets. Our numerical computations are based on both maximum a posteriori and
conditional mean estimation.We exploit state-of-the-art MCMC methodologies such
as Metropolis-within-Gibbs, Repelling-Attracting Metropolis, and No-U-Turn
sampler variant of Hamiltonian Monte Carlo. We demonstrate the models and
methods constructed for one-dimensional and two-dimensional deconvolution
problems. Thorough MCMC statistics are provided for all test cases, including
potential scale reduction factors.
