基于高斯尺度混合的带Student's t先验的贝叶斯反演
Bayesian inversion with Student's t priors based on Gaussian scale mixtures
AI总结：
针对贝叶斯反问题的结构先验建模需求，提出结合马尔可夫随机场与Student's t分布的先验类，借助高斯尺度混合表示实现高效后验计算，经信号去卷积和图像去模糊算例验证有效性。
摘要：
许多反问题聚焦于复原先验已知具有不连续或光滑特性的目标量。在贝叶斯反问题方法中，这类结构信息可通过马尔可夫随机场先验进行编码。我们提出一类结合马尔可夫随机场结构与Student's t分布的先验。该方法可根据现有数据灵活建模多样的结构特性，其灵活性通过将Student's t分布的自由度参数纳入贝叶斯反问题公式实现。为便于后验计算，我们对Student's t马尔可夫随机场先验采用高斯尺度混合表示，该表示可将先验表达为依赖辅助超参数的条件高斯分布。采用该表示后，我们可推导得到多数后验条件分布的解析形式，并利用Gibbs采样器探索后验分布。我们通过两个数值算例——信号去卷积与图像去模糊——验证了该方法。
英文摘要：
Many inverse problems focus on recovering a quantity of interest that is a priori known to exhibit either discontinuous or smooth behavior. Within the Bayesian approach to inverse problems, such structural information can be encoded using Markov random field priors. We propose a class of priors that combine Markov random field structure with Student's t distribution. This approach offers flexibility in modeling diverse structural behaviors depending on available data. Flexibility is achieved by including the degrees of freedom parameter of Student's t distribution in the formulation of the Bayesian inverse problem. To facilitate posterior computations, we employ Gaussian scale mixture representation for the Student's t Markov random field prior, which allows expressing the prior as a conditionally Gaussian distribution depending on auxiliary hyperparameters. Adopting this representation, we can derive most of the posterior conditional distributions in a closed form and utilize the Gibbs sampler to explore the posterior. We illustrate the method with two numerical examples: signal deconvolution and image deblurring.
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