  Global-local shrinkage hierarchies are an important innovation in Bayesian
estimation. We propose the use of log-scale distributions as a novel basis for
generating familes of prior distributions for local shrinkage hyperparameters.
By varying the scale parameter one may vary the degree to which the prior
distribution promotes sparsity in the coefficient estimates. By examining the
class of distributions over the logarithm of the local shrinkage parameter that
have log-linear, or sub-log-linear tails, we show that many standard prior
distributions for local shrinkage parameters can be unified in terms of the
tail behaviour and concentration properties of their corresponding marginal
distributions over the coefficients $\beta_j$. We derive upper bounds on the
rate of concentration around $|\beta_j|=0$, and the tail decay as $|\beta_j|
\to \infty$, achievable by this wide class of prior distributions.
  We then propose a new type of ultra-heavy tailed prior, called the log-$t$
prior with the property that, irrespective of the choice of associated scale
parameter, the marginal distribution always diverges at $\beta_j = 0$, and
always possesses super-Cauchy tails. We develop results demonstrating when
prior distributions with (sub)-log-linear tails attain Kullback--Leibler
super-efficiency and prove that the log-$t$ prior distribution is always
super-efficient. We show that the log-$t$ prior is less sensitive to
misspecification of the global shrinkage parameter than the horseshoe or lasso
priors. By incorporating the scale parameter of the log-scale prior
distributions into the Bayesian hierarchy we derive novel adaptive shrinkage
procedures. Simulations show that the adaptive log-$t$ procedure appears to
always perform well, irrespective of the level of sparsity or signal-to-noise
ratio of the underlying model.
