Just to be super clear, this experiment definitely isn’t definitive on the topic, but it was enough to convince me that it wasn’t worth going further. A more detail-oriented person would’ve done a simulation study with known parameters and checked bias and coverage :p
The problems with the funnel is a problem with only using first-order information. @betanalpha did some super-cool calculations that showed that the re-parameterization is equivalent to running Riemannian Manifold HMC with a specific, non-constant mass matrix. So this suggests to me that the more principled fix to the geometry problem (compared to changing the model) is to use second-order information in the sampler. To change the model without modelling reasons to do it just feels like the tail wagging the dog.
Edit (so not to spam the thread): My experiments used a Gamma (which goes to zero slower than a lognormal near zero) and an inverse Gamma (which goes to zero much faster than the log-normal near zero). They both failed but it’s possible that a log-normal would do better so I checked. here is the equivalent table.
In all these experiments, the lognormal prior has mean given in the first column (the lower levels are quite silly!) and standard deviation 1. The code is in the git repo if you want to try some more values of the standard deviation (there might be a magic combo that makes sense, but I’m pretty lazy).
I’m seeing the same sort of behaviour: the values priors that remove the geometry problem massively push the estimates of the group variance to the right. For comparison, the values of \tau you get with a half-N(0,5) prior on the standard deviation (and a non-centred parameterization) are (0.13, 2.69, 9.23) [(0.025,0.5,0.975)-quanitles].
| mu | prior median | 0.01 quantile | divergences | low_BFMI | 2.5% for tau | 50% for tau | 97.5% for tau |
|---|---|---|---|---|---|---|---|
| 0.00 | 1.00 | 0.10 | 170 | FALSE | 0.34 | 1.16 | 5.57 |
| 0.50 | 1.65 | 0.16 | 497 | TRUE | 0.38 | 1.57 | 7.69 |
| 1.00 | 2.72 | 0.27 | 179 | FALSE | 0.81 | 2.55 | 10.58 |
| 1.50 | 4.48 | 0.44 | 149 | FALSE | 1.03 | 3.27 | 11.46 |
| 2.00 | 7.39 | 0.72 | 57 | FALSE | 1.11 | 4.43 | 14.33 |
| 2.50 | 12.18 | 1.19 | 69 | FALSE | 1.12 | 5.04 | 16.36 |
| 3.00 | 20.09 | 1.96 | 16 | FALSE | 1.92 | 6.67 | 18.85 |
| 4.00 | 54.60 | 5.33 | 10 | FALSE | 2.39 | 8.96 | 23.65 |
| 5.00 | 148.41 | 14.49 | 1 | FALSE | 4.23 | 11.86 | 30.65 |
| 6.00 | 403.43 | 39.40 | 1 | FALSE | 5.94 | 15.22 | 38.94 |
| 7.00 | 1096.63 | 107.09 | 0 | FALSE | 7.72 | 18.86 | 48.33 |
| 8.00 | 2980.96 | 291.10 | 0 | FALSE | 9.62 | 23.68 | 68.11 |
| 9.00 | 8103.08 | 791.28 | 0 | FALSE | 12.01 | 30.88 | 91.27 |
| 10.00 | 22026.47 | 2150.92 | 0 | FALSE | 14.46 | 40.34 | 145.52 |