Can the horseshoe prior be used with Beta(b, b) instead of Beta(.5, .5)?

Question

The horseshoe prior for, say, an unknown mean $\mu$ is expressed $$ \begin{align*} \mu_i | \lambda_i, \tau &\sim N(0, \lambda_i^2 \tau^2) \\ \lambda_i &\sim C^+(0,1) \end{align*} $$ where the prior on $\lambda_i$ can be equivalently stated as $\kappa_i := \frac{1}{1+\lambda_i^2} \sim$ Beta(.5,.5), and it is this distribution on $\kappa_i$ that has the horseshoe shape for which the scheme is named. In the normal mean model $y | \mu \sim N_d (\mu, I)$, the $\kappa_i$ say how much the MLE $y$ should be shrunk, and the philosophy of the horseshoe is that you want $\kappa_i$ to be near 0 to produce no shrinkage in signal components, or near 1 to produce strong shrinkage to 0 of noise components. Hence you'd like a prior on $\kappa_i$ that has its mass toward 0 and 1, like Beta(.5,.5).

However, Beta(b,b) for b in (0,1) will have the same horseshoe shape. $b$ approaching 0 will shift mass out of the middle toward 0 and 1, with limiting distribution Bernoulli(.5). $b$ approaching 1 will shift mass from the ends into the middle, approaching U(0,1).

Is there any work on using values of $b$ other than .5 in horseshoe priors? If so, what guides such choices?

Answer 1

In the subjective Bayesian paradigm, you always have a choice of priors, so obviously you can use that alternative prior if you want. The only real issue here is whether or not you call that other prior a "horseshoe", which is really an aesthetic/terminology judgment, not a substantive statistical issue.$^\dagger$

As to what would guide the use of other types of beta priors with $0<b<1$ (compared to using $b = \tfrac{1}{2}$), presumably we would use $\tfrac{1}{2}<b<1$ when we want to be closer to a uniform distribution, and we would use a value $0<b<\tfrac{1}{2}$ when we want greater prior mass at the extremities. In the context of the model you mention for signals, this would be a contextual judgment of how much we wish to favour values near zero or one a priori.

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$^\dagger$ Incidentally, a rather silly aspect of the whole "horseshoe" terminology is the implicit assumption that the horseshoe can only be looked at in one direction (i.e., held so that it appears convex). If you rotate the horseshoe 180-degrees it will then appear concave, so we could just as easily call concave priors "horseshoe" priors. In fact, none of these priors actually look like horseshoes, since actual horseshoes rotate around to an extent that they do not even look like a function.