What if you like the flavours of both Bayesian inference and the implicit model selection of sparse inference? Can you cook Bayesian-Frequentist fusion cuisine with this novelty ingredient?
Laplace Prior
Laplace priors on linear regression coefficients, includes normal lasso as a MAP estimate.
Pro: It is easy to derive frequentist LASSO as a MAP estimate from this prior.
Con: Not actually sparse for non-MAP uses.
I have no need for this right now, but I did I might start with Dan Simpson’s critique.
Spike-and-slab prior
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Horseshoe prior
Stan guy, Michael Betancourt introduces some issues with LASSO-type inference for Bayesians with a slant towards Horseshoe-type priors in preference spike and slab, possibly because hierarchical mixtures like spike-and-slab are not that great in Stan, albeit possible.
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———. 2010. “The Horseshoe Estimator for Sparse Signals.” Biometrika 97 (2): 465–80. https://doi.org/10.1093/biomet/asq017.
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