Working with Matthew Stephens and Nicholas Polson, I’m exploring various ideas in statistical shrinkage, selection, and sparsity, especially in Bayesian framework.

• Bernoulli-Gaussian spike-and-slab applied to normal means
• Bernoulli-Laplace spike-and-slab applied to normal means

The problem is whether we can find a $\phi$, such that $\hat\mu_B$, the optimal Bayesian estimator to a certain loss, is a solution to the regularized least squares with $\phi$ as the penalty. This framework of matching Tweedie’s formula to a proximal operator can potentially be generalized to the exponential family likelihood, not just normal means. The specific formula should be changed accordingly.

• Single best replacement (SBR) for $l_0$-regularized linear regression: Accuracy
• Single best replacement (SBR) for $l_0$-regularized linear regression: Time 1
• Single best replacement (SBR) for $l_0$-regularized linear regression: Time 2
• Single best replacement (SBR) for $l_0$-regularized linear regression: Diabetes data set

$l_0$-regularized linear regression is NP-hard, yet under high SNR and high collinearity, the single best replacement (SBR) algorithm, developed in the signal processing community, is compared favorably to $l_1$ methods like lasso, elastic net, $l_q, q\in(0, 1)$ method like BayesBridge, and the gold standard spike-and-slab MCMC.

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