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 Simpsonβs critique.
Spike-and-slab prior
π
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.
References
Babacan, S. Derin, Martin Luessi, Rafael Molina, and Aggelos K. Katsaggelos. 2012. βSparse Bayesian Methods for Low-Rank Matrix Estimation.β IEEE Transactions on Signal Processing 60 (8): 3964β77.
Bondell, Howard D., and Brian J. Reich. 2012. βConsistent High-Dimensional Bayesian Variable Selection via Penalized Credible Regions.β Journal of the American Statistical Association 107 (500): 1610β24.
Brodersen, Kay H., Fabian Gallusser, Jim Koehler, Nicolas Remy, and Steven L. Scott. 2015. βInferring Causal Impact Using Bayesian Structural Time-Series Models.β The Annals of Applied Statistics 9 (1): 247β74.
Carvalho, Carlos M., Nicholas G. Polson, and James G. Scott. 2009. βHandling Sparsity via the Horseshoe.β In Artificial Intelligence and Statistics, 73β80.
βββ. 2010. βThe Horseshoe Estimator for Sparse Signals.β Biometrika 97 (2): 465β80.
George, Edward I., and Robert McCulloch. 1997. βApproaches for bayesian variable selection.β Statistica Sinica 7 (2): 339β73.
Ishwaran, Hemant, and J. Sunil Rao. 2005. βSpike and Slab Variable Selection: Frequentist and Bayesian Strategies.β The Annals of Statistics 33 (2): 730β73.
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Piironen, Juho, and Aki Vehtari. 2017. βSparsity Information and Regularization in the Horseshoe and Other Shrinkage Priors.β Electronic Journal of Statistics 11 (2): 5018β51.
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Schniter, Philip, Lee C. Potter, and Justin Ziniel. 2008. βFast Bayesian Matching Pursuit.β In 2008 Information Theory and Applications Workshop, 326β33. San Diego, CA, USA: IEEE.
Scott, Steven L., and Hal R. Varian. 2013. βPredicting the Present with Bayesian Structural Time Series.β SSRN Scholarly Paper ID 2304426. Rochester, NY: Social Science Research Network.
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Titsias, Michalis K., and Miguel LΓ‘zaro-Gredilla. 2011. βSpike and Slab Variational Inference for Multi-Task and Multiple Kernel Learning.β In Advances in Neural Information Processing Systems 24, edited by J. Shawe-Taylor, R. S. Zemel, P. L. Bartlett, F. Pereira, and K. Q. Weinberger, 2339β47. Curran Associates, Inc.
Zhou, Mingyuan, Haojun Chen, John Paisley, Lu Ren, Guillermo Sapiro, and Lawrence Carin. 2009. βNon-Parametric Bayesian Dictionary Learning for Sparse Image Representations.β In Proceedings of the 22nd International Conference on Neural Information Processing Systems, 22:2295β2303. NIPSβ09. Red Hook, NY, USA: Curran Associates Inc.
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