How to make the regularized horseshoe less strict?


we use the regularized horseshoe for variable selection. If I estimate our model without the reg. horseshoe prior considering only a small subset of 10 predictors there are significant effects on the 5% level (i.e. 95%-CI does not contain zero) as well as 10% level (i.e. 90%-CI does not contain zero). Now we have the problem that by using the reg. horseshoe the effects of the predictors on the 5% level are still there, but all 90% CI from the non-horseshoe model now contain zero, but we still want to allow for these effects.

We now wonder if there is any way to tweak the prior in a way, that it also allows for effects that are not that far away from zero. I already tried to set the global scale for the hyper prior larger, I also set \tau manually to a larger value. But the results didn’t change. Is there any other way make the reg. horseshoe less strict in terms of shrinkage?

Thank you in advance for your ideas!

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Is the data simulated from a known process?
Is the regularized horseshoe used also for 10 predictors?
Are the predictions and the predictive performance what you would expect?