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 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 Matérn 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.

在统计逆问题中使用柯西马尔可夫随机场先验可能会导致非高斯、高维、多峰和重尾的后验分布。为了成功使用这些先验，通常需要复杂的优化和马尔可夫链蒙特卡罗方法。在本文中，我们的重点主要是回顾最近开发的 Cauchy 差异先验，同时引入有趣的新变体，同时提供比较。我们首先提出一维二阶柯西差分先验，并构造新的一阶和二阶二维各向同性柯西差分先验。另一个新的 Cauchy 先验基于随机偏微分方程方法，源自 Matérn 型高斯表示。比较还包括柯西表。我们的数值计算基于最大后验和条件均值估计。我们利用最先进的 MCMC 方法，例如 Metropolis-within-Gibbs、Repelling-Attracting Metropolis 和 Hamiltonian Monte Carlo 的 No-U-Turn sampler 变体。我们展示了为一维和二维反卷积问题构建的模型和方法。为所有测试用例提供了完整的 MCMC 统计数据，包括潜在的缩减因子。