Bayesian estimation of sparse signals with a continuous spike-and-slab prior
The author provides a unified perspective on Bayesian estimation of sparse signals with continuous spike-and-slab prior combining both penalized likelihood and fully Bayes perspective. More precisely, the author focuses on the canonical problem of estimation a high dimensional mean vector from a single multivariate observation. The observed vector arises from \[ Y_i=\beta_{0i}+\epsilon_i,\quad \epsilon\sim N(0,1),\;i=1,\dots,n, \] and the goal is to estimate \(\beta=(\beta_{01,}, \dots, \beta_{0n})^{\prime}\) under squared error loss. The sparsity is defined as \(\beta_0\in l_0[p_n;n]= \{\beta \in \mathbb{R}^n\,:\,\sum_{i=1}^n I(|\beta_i|\neq0)\leq p_n\}\) and \(p(n)=o(n)\) as \(n\to\infty\). The quality of recovery is assessed relative to the benchmark minimax risk \(2p_n\log (n/p)(1+o(1))\) and the mean minimax risk \(2p_n\log\,n(1+o(1))\). The author adapts the view that a good point estimator should have a minimax risk that is always within a universal constant multiple of the (near) minimax risk. The spike-and-slab prior SS-LASSO prior is of the form \[ \pi(\beta\,|\,\gamma)=\prod_{i=1}^{n}[(1-\gamma_i)\psi (\beta_i\,|\,\lambda_0)+\gamma_i \psi(\beta_i\,|\,\lambda_1)], \quad \gamma\sim\pi(\phi\,|\,\theta), \] where \(\psi(\beta\,|\,l)=(\lambda/2)\exp\{-\lambda|\beta|\}\) is a Laplace distribution with mean 0 and variance \(2/\lambda^2\). For the class of SS-LASSO priors a rigorous frequentist analysis of the entire posterior distribution and its modes is provided.
scientific article; zbMATH DE number 6865116
scientific article