论文标题
贝叶斯分层模型中最大后验估计器的路径遵循方法：估计如何取决于超参数
Path-following methods for Maximum a Posteriori estimators in Bayesian hierarchical models: How estimates depend on hyperparameters
论文作者
论文摘要
与所有贝叶斯方法一样，最大后验（MAP）估计取决于先前的假设。通常选择这些假设以促进恢复估计值中的特定特征。选定的先验形式决定了后验分布的形状，因此估计值的行为和相关优化问题的复杂性。在这里，我们考虑了一个高斯层次模型的家族，其旨在促进线性反问题的稀疏性具有广义的伽玛高度培训。通过改变超参数，我们在Pross flooke $ \ ell_p $罚款的先验之间连续移动，并使用灵活的$ p $，平滑和比例。然后，我们引入了一种预测器 - 矫正器方法，该方法跟踪MAP解决方案路径作为超参数变化。路径以下使用户可以探索可能的MAP解决方案的空间，并测试解决方案对先前假设更改的敏感性。通过追踪从凸区域到非凸区域的路径，用户可以在强烈的稀疏性中找到局部最小化器，从而促进与使用相关先前假设得出的凸宽松相一致的凸状态。我们通过实验表明这些解决方案。与非凸问题的直接优化相比，误差易于错误。
Maximum a posteriori (MAP) estimation, like all Bayesian methods, depends on prior assumptions. These assumptions are often chosen to promote specific features in the recovered estimate. The form of the chosen prior determines the shape of the posterior distribution, thus the behavior of the estimator and complexity of the associated optimization problem. Here, we consider a family of Gaussian hierarchical models with generalized gamma hyperpriors designed to promote sparsity in linear inverse problems. By varying the hyperparameters, we move continuously between priors that act as smoothed $\ell_p$ penalties with flexible $p$, smoothing, and scale. We then introduce a predictor-corrector method that tracks MAP solution paths as the hyperparameters vary. Path following allows a user to explore the space of possible MAP solutions and to test the sensitivity of solutions to changes in the prior assumptions. By tracing paths from a convex region to a non-convex region, the user can find local minimizers in strongly sparsity promoting regimes that are consistent with a convex relaxation derived using related prior assumptions. We show experimentally that these solutions. are less error prone than direct optimization of the non-convex problem.
