
230 Current Trends in Bayesian Methodology with Applications
−3 −2 −1
0 1
2 3
β
π(β)
GDP
Cauchy
Laplace
HS
Optimal
4.0 4.5 5.0 5.5 6.0 6.5 7.0
0.0 0.2 0.4 0.6 0.8
0.00 0.01 0.02 0.03 0.04
β
π(β)
GDP
Cauchy
Laplace
HS
Optimal
(a) Shrinkage near zero (b) Tail behaviour
FIGURE 11.1
Probability density functions for generalized double Pareto, Cauchy, Laplace,
Horseshoe and Robust prior.
n i.i.d. N(0, σ
2
) r.v.s. For convenience in presentation, σ
2
is assumed known.
The parameter dimension p may be larger than n, i.e. p > n in which ca se we
have a high-dimensional problem, but we may also have p < n. Various details
in the Lasso based analysis dep e nd o n whether p > n or