Different horseshoe specifications

Dear all,

In reading the literature on the horseshoe and also scanning the Stan forum, I’m coming across a couple of different specifications for the ordinary horseshoe (I know there is a Finnish version). For example, in Carvalho, Polson, and Scott (2009) we find

\beta_{m} \sim \mathcal{N} (0, \tau \cdot \lambda_{m})

\lambda_{m} \sim \text{Half-}\mathcal{C} (0, 1)

\tau \sim \text{Half-}\mathcal{C} (0, 1).

but in Piironen and Vehtari (2017) we find that \tau and \lambda are both squared.

Then in this post (Horseshoe prior in rstan) the gentleman multiples \tau and \lambda by \sigma.

I’m interested in simply knowing the proper specification of the horseshoe prior. Right now, I’m using the Carvalho, Polson, and Scott (2009) which seems to work fine.

Thank you!

@jpiironen @avehtari

There is no one single proper specification, and different versions can work similalrly or differently depending on the case. See some discussion in sections 3.2, 3.3 and 3.5, and appendices C.1 and C.2 in

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