x < 1, a > 0, b > 0 and ϕ > 0. When ϕ = 1, the TPB distribution becomes a standard Beta distribution. For a fixed value of ϕ, a controls the behavior of the TPB prior near one, and thus the behavior of the prior on βj around zero; b controls the behavior of the TPB prior near zero, and thus affects the tails of the prior on βj. Figure 3 shows the prior densities on τj (upper panel) and βj (middle and lower panels) with ϕ = 1, b = 1/2, and three different values of a: a = 1/2, a = 1 and a = 3/2. It can be seen that when a = 1/2 and b = 1/2, the TPB prior has substantial mass near zero and one (Fig. 3, upper panel), and thus the corresponding prior density on βj has a very sharp peak around the origin, with zero being a pole (singular point; Fig. 3, middle panel), along with heavy, Cauchy-like tails (Fig. 3, lower panel). This prior is known as the horseshoe prior50, due to the horseshoe-shaped prior density on the shrinkage factor τj. As a increases, the prior on βj
