Enes Makalic

Shu Yu Tew, Daniel F. Schmidt, Enes Makalic

The horseshoe prior is known to possess many desirable properties for Bayesian estimation of sparse parameter vectors, yet its density function lacks an analytic form. As such, it is challenging to find a closed-form solution for the posterior mode. Conventional horseshoe estimators use the posterior mean to estimate the parameters, but these estimates are not sparse. We propose a novel expectation-maximisation (EM) procedure for computing the MAP estimates of the parameters in the case of the standard linear model. A particular strength of our approach is that the M-step depends only on the form of the prior and it is independent of the form of the likelihood. We introduce several simple modifications of this EM procedure that allow for straightforward extension to generalised linear models. In experiments performed on simulated and real data, our approach performs comparable, or superior to, state-of-the-art sparse estimation methods in terms of statistical performance and computational cost.

Daniel F. Schmidt, Enes Makalic

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 $β_j$. We derive upper bounds on the rate of concentration around $|β_j|=0$, and the tail decay as $|β_j| \to \infty$, 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 $β_j = 0$, 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.