We propose to use L\'evy {\alpha}-stable distributions for constructing priors for Bayesian inverse problems. The construction is based on Markov fields with stable-distributed increments. Special cases include the Cauchy and Gaussian distributions, with stability indices {\alpha} = 1, and {\alpha} = 2, respectively. Our target is to show that these priors provide a rich class of priors for modelling rough features. The main technical issue is that the {\alpha}-stable probability density functions do not have closed-form expressions in general, and this limits their applicability. For practical purposes, we need to approximate probability density functions through numerical integration or series expansions. Current available approximation methods are either too time-consuming or do not function within the range of stability and radius arguments needed in Bayesian inversion. To address the issue, we propose a new hybrid approximation method for symmetric univariate and bivariate {\alpha}-stable distributions, which is both fast to evaluate, and accurate enough from a practical viewpoint. Then we use approximation method in the numerical implementation of {\alpha}-stable random field priors. We demonstrate the applicability of the constructed priors on selected Bayesian inverse problems which include the deconvolution problem, and the inversion of a function governed by an elliptic partial differential equation. We also demonstrate hierarchical {\alpha}-stable priors in the one-dimensional deconvolution problem. We employ maximum-a-posterior-based estimation at all the numerical examples. To that end, we exploit the limited-memory BFGS and its bounded variant for the estimator.

翻译：我们提出使用Lévy α稳定分布为贝叶斯逆问题构造先验分布。该构造基于具有稳定分布增量的马尔可夫场，其特例包括柯西分布（稳定性指数α=1）和高斯分布（α=2）。旨在证明此类先验能为粗糙特征建模提供丰富的先验类别。主要技术难点在于α稳定概率密度函数通常不存在闭式表达式，这限制了其应用性。在实际应用中，需通过数值积分或级数展开近似概率密度函数，但现有近似方法要么计算代价过高，要么无法满足贝叶斯反演所需的稳定参数与尺度参数范围。为此，我们提出一种新型混合近似方法，针对对称单变量和双变量α稳定分布，既能快速求值又能满足实际精度需求。随后将该近似方法应用于α稳定随机场先验的数值实现中。我们通过选定的贝叶斯逆问题（包括反卷积问题和椭圆偏微分方程控制函数反演问题）验证所构建先验的适用性，并在二维反卷积问题中展示了分层α稳定先验的应用。所有数值示例均采用最大后验估计，并利用有限内存BFGS算法及其边界约束变体进行估计。