L1-ball-type priors are a recent generalization of the spike-and-slab priors. By transforming a continuous precursor distribution to the L1-ball boundary, it induces exact zeros with positive prior and posterior probabilities. With great flexibility in choosing the precursor and threshold distributions, we can easily specify models under structured sparsity, such as those with dependent probability for zeros and smoothness among the non-zeros. Motivated to significantly accelerate the posterior computation, we propose a new data augmentation that leads to a fast block Gibbs sampling algorithm. The latent variable, named ``anti-correlation Gaussian'', cancels out the quadratic exponent term in the latent Gaussian distribution, making the parameters of interest conditionally independent so that they can be updated in a block. Compared to existing algorithms such as the No-U-Turn sampler, the new blocked Gibbs sampler has a very low computing cost per iteration and shows rapid mixing of Markov chains. We establish the geometric ergodicity guarantee of the algorithm in linear models. Further, we show useful extensions of our algorithm for posterior estimation of general latent Gaussian models, such as those involving multivariate truncated Gaussian or latent Gaussian process. Keywords: Blocked Gibbs sampler; Fast Mixing of Markov Chains; Latent Gaussian Models; Soft-thresholding.

翻译：L1球型先验是尖峰-平板先验的最新推广。通过将连续前体分布变换到L1球边界，该方法能以正先验概率和后验概率诱导出精确零点。由于在选择前体分布和阈值分布方面具有高度灵活性，我们可以轻松指定结构化稀疏性下的模型，例如具有零点依赖概率和非零点光滑性的模型。为显著加速后验计算，我们提出一种新的数据增强方法，从而得到快速块吉布斯采样算法。该潜变量被命名为"反相关高斯"，它消除了潜高斯分布中的二次指数项，使感兴趣参数条件独立，从而可以按块更新。与No-U-Turn采样器等现有算法相比，新提出的块吉布斯采样器每次迭代的计算成本极低，且马尔可夫链混合速度极快。我们在线性模型中建立了算法的几何遍历性保证。此外，我们还展示了该算法在一般潜高斯模型（例如涉及多元截断高斯或潜高斯过程的模型）后验估计中的有效扩展。关键词：块吉布斯采样器；马尔可夫链快速混合；潜高斯模型；软阈值。