6/7/2019, Andrew Womack, Indiana University–Bloomington Horseshoes with heavy tails
From Kathie Leck
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Horseshoes with heavy tails
In high dimensional problems, the usual two groups problem of model selection is impossible due to the combinatorial complexity of the model space. In recent years, a set of one group models that approximates the two groups problem have been developed. Of these, the Horseshoe prior is perhaps the most famous and places a Beta(1/2,1/2) prior on the local shrinkage parameters.
There are many modifications and extensions of this framework, and we propose a new modification. Specifically, we model the local shrinkage parameter as a Beta(p,1-p) for each parameter under consideration in order to mimic the model selection problem. Placing priors on the p produces a prior distribution with extremely heavy tails that yields both very strong shrinkage of small signals and unbiased estimation of large signals, having overall better risk behavior. We also consider other prior specifications for p that provide superior inference in super-sparse settings.