  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 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 testing procedure of whether or not a Laplace prior 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 priors which correspond to $\ell_q$ 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
numbers of observations and unknown regression coefficients are large. We then
introduce an adaptive procedure that chooses a more appropriate prior and
corresponding penalty from the class of exponential power priors when the null
hypothesis is rejected. We show that this can improve estimation of the
regression coefficients both when they are drawn from an exponential power
distribution and when they are drawn from a spike-and-slab distribution.
