I have seen that the regularized horseshoe has been offered as a continuous counterpart of a spike and slab. However, is there any continous counterparts to a spike and slab with a slab that is not centred at zero?

Any discrete spike-and-slab prior can be approximated by a mixture of continuous priors, specifically,

\pi_{\text{spike}}(\beta | \mu_{\text{spike}}) = \delta_{\mu_\text{spike}}(\beta) \approx N(\beta | \mu_{\text{spike}}, \epsilon^2)

for some small \epsilon.

It follows that

\pi_{ss}(\beta) = \gamma \pi_{\text{slab}}(\beta) + (1 - \gamma) \delta_{\mu_\text{spike}}(\beta) \approx \gamma \pi_{\text{slab}}(\beta) + (1 - \gamma) N(\beta | \mu_{\text{spike}}, \epsilon^2).

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`rstanarm`

and `brms`

have the slab mean fixed at 0, but if you write your model in Stan language, you can use the code example in the Appendix C of the paper Sparsity information and regularization in the horseshoe and other shrinkage priors, and set the slab mean to be non-zero.

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