Bayesian sparsity

2019-01-08 — 2024-11-19

Bayes
high d
information
model selection
regression
sparser than thou
statistics

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What if I like the flavours of both Bayesian inference and the implicit model selection of sparse inference? Can I cook Bayesian-Frequentist fusion cuisine with this novelty ingredient? It turns out that yes, we can add a variety of imitation sparsity flavours to Bayes model selection. The resulting methods are handy in, for example, symbolic system identification.

Figure 1

1 Laplace Prior

Laplace priors on linear regression coefficients include normal lasso as a MAP estimate.

Pro: It is easy to derive frequentist LASSO as a MAP estimate from this prior.

Con: Full posterior is not sparse, only the MAP estimate.

I have no need for this right now, but if I did, I might start with Dan Simpson’s critique.

2 Spike-and-slab prior

Probabilistic equivalent of a LASSO-type prior, via a hierarchical mixture, i.e. an inclusion probability for each coefficient.

Pro: Full posterior can be sparse in some sense.

Con: Mixtures of discrete and continuous variables like this are fiddly to deal with in MCMC, and just in general.

🏗

3 Horseshoe prior

Figure 2

Stan guy, Michael Betancourt introduces some issues with LASSO-type inference for Bayesians with a slant towards Horseshoe-type priors in preference to spike-and-slab, possibly because hierarchical mixtures like spike-and-slab are not that easy in Stan, albeit possible.

4 Thompson sampling for large model spaces

This looks cool: Liu and Ročková ().

5 Transdimensional inference

See transdimensional.

6 Global-local shrinkage hierarchy

Not quite sure what this is, but here are some papers: Bhadra et al. ();Polson and Scott ();Schmidt and Makalic ();Xu et al. ().

7 As Singular Learning

Sparse bayes problems are clearly connecting to singular learning theory approaches to Bayes. TBD

8 Incoming

9 References

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———. 2010. “The Horseshoe Estimator for Sparse Signals.” Biometrika.
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