Shrinkage with Shrunken Shoulders: Gibbs Sampling Shrinkage Model Posteriors with Guaranteed Convergence Rates
Shrinkage with Shrunken Shoulders: Gibbs Sampling Shrinkage Model Posteriors with Guaranteed Convergence Rates
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DOI：
10.1214/22-ba1308
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发表时间：
2023-06-01
影响因子：
4.4
通讯作者：
Suchard, Marc A.
中科院分区：
文献类型：
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作者：
Nishimura, Akihiko;Suchard, Marc A.
使用连续收缩先验-具有接近零的“尖峰”和朝向无穷大的重尾-是一种越来越流行的方法，以在参数估计中引入稀疏性。然而，当参数仅通过似然性弱识别时，后验结果可能会像前验结果一样具有较重的尾部，从而危及推理的鲁棒性。一个自然的解决方案是通过在合理的参数范围之外减轻其尾部来“收缩收缩先验的肩部”，从而产生先验的正则化版本。我们开发了一种正则化方法，与以前的建议不同，它保留了原始收缩先验的计算上有吸引力的结构。我们研究了Gibbs抽样器在后验分布上的理论性质，重点研究了稀疏逻辑回归的Po ' lya-Gamma Gibbs抽样器的收敛速度。我们的分析表明，提出的正则化导致几何遍历下广泛的全球局部收缩先验。本质上，唯一的要求是局部尺度A上的先验rlocal（中心点）满足rlocal（0）<00。如果rlocal（中心点）进一步满足lim lambda -> 0 rlocal（A）/Aa < oo for a >0，如贝叶斯桥先验的情况下，我们证明了采样器是一致遍历的。
Use of continuous shrinkage priors - with a "spike" near zero and heavy-tails towards infinity - is an increasingly popular approach to induce spar-sity in parameter estimates. When the parameters are only weakly identified by the likelihood, however, the posterior may end up with tails as heavy as the prior, jeopardizing robustness of inference. A natural solution is to "shrink the shoulders" of a shrinkage prior by lightening up its tails beyond a reasonable parameter range, yielding a regularized version of the prior. We develop a regularization approach which, unlike previous proposals, preserves computationally attractive structures of original shrinkage priors. We study theoretical properties of the Gibbs sam-pler on resulting posterior distributions, with emphasis on convergence rates of the Po ' lya-Gamma Gibbs sampler for sparse logistic regression. Our analysis shows that the proposed regularization leads to geometric ergodicity under a broad range of global-local shrinkage priors. Essentially, the only requirement is for the prior rlocal(center dot) on the local scale A to satisfy rlocal(0) < oo. If rlocal(center dot) further satisfies lim lambda -> 0 rlocal(A)/Aa < oo for a > 0, as in the case of Bayesian bridge priors, we show the sampler to be uniformly ergodic.