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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, itinduces exact zeros with positive prior and posterior probabilities. With greatflexibility in choosing the precursor and threshold distributions, we caneasily specify models under structured sparsity, such as those with dependentprobability for zeros and smoothness among the non-zeros. Motivated tosignificantly accelerate the posterior computation, we propose a new dataaugmentation that leads to a fast block Gibbs sampling algorithm. The latentvariable, named ``anti-correlation Gaussian'', cancels out the quadraticexponent term in the latent Gaussian distribution, making the parameters ofinterest conditionally independent so that they can be updated in a block.Compared to existing algorithms such as the No-U-Turn sampler, the new blockedGibbs sampler has a very low computing cost per iteration and shows rapidmixing of Markov chains. We establish the geometric ergodicity guarantee of thealgorithm in linear models. Further, we show useful extensions of our algorithmfor posterior estimation of general latent Gaussian models, such as thoseinvolving multivariate truncated Gaussian or latent Gaussian process. Keywords:Blocked Gibbs sampler; Fast Mixing of Markov Chains; Latent Gaussian Models;Soft-thresholding.

Gibbs Sampling using Anti-correlation Gaussian Data Augmentation, with Applications to L1-ball-type Models