Testing Sparsity-Inducing Penalties
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It is well understood that many penalized maximum likelihood estimators correspond to posterior mode estimators under specific prior distributions. Appropriateness of a particular class of penalty functions can therefore be interpreted as the appropriateness of a prior model for the parameters. For example, the appropriateness of a lasso penalty for regression coefficients depends on the extent to which the empirical distribution of the regression coefficients resembles a Laplace distribution. We give a simple approximate testing procedure of whether or not a Laplace prior model is appropriate and accordingly, whether or not using a lasso penalized estimate is appropriate. This testing procedure is designed to have power against exponential power prior models which correspond to ℓ_q penalties. Via simulations, we show that this testing procedure achieves the desired level and has enough power to detect violations of the Laplace assumption when the number of observations and number of unknown regression coefficients are large. We then introduce an adaptive procedure that chooses a more appropriate prior model and corresponding penalty from the class of exponential power prior models when the null hypothesis is rejected. We show that this computationally simple adaptive procedure can improve estimation of the unknown regression coefficients both when the unknown regression coefficients are drawn from an exponential power distribution and when the unknown regression coefficients are sparse and drawn from a spike-and-slab distribution.
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