NUTS / Funnel Lab
See why NUTS can fix path-length tuning and still fail on bad hierarchical geometry. The real choice is often centered vs non-centered coordinates, not sampler branding.
NUTS / Funnel Lab
Geometry decides whether warmup can save the run
Watch the same hierarchical model in two coordinate systems. The centered form can squeeze into a sharp funnel neck; the non-centered form spreads that geometry back onto a friendlier latent scale. NUTS adapts path length, but it still inherits the geometry you hand it.
Blue or green points are posterior mass, the dashed path is the adapted trajectory, and rose crosses mark where divergences cluster first.
Centered hierarchy
Local effects are stored directly, so weak data forces the sampler through a narrow scale-dependent neck.
Bulk ESS / 1k draws
211
What to watch
Neck pinches, path kinks, and divergences cluster where the local scale collapses.
Non-centered hierarchy
Local effects move onto a standard-normal base scale, so the same posterior becomes much easier for HMC and NUTS to traverse.
Bulk ESS / 1k draws
465
What to watch
Cloud stays close to isotropic and the adapted path spends more time exploring than recovering.
Low strength means the shared scale parameter dominates. High strength means each group effect is pinned down by its own likelihood.
This bundles step size ambition and how hard the sampler tries to move through the posterior in one trajectory. High values reveal bad geometry faster.
Try this
Start with the weak-data funnel preset. Then keep the same weak data regime and reduce aggressiveness. You should see tuning help a little, but the non-centered panel remains much easier because the real fix is geometric.
Scenario note
Centered geometry pinches into a narrow neck, so aggressive integration throws divergences there first.
Diagnosis
Classic weak-data funnel
The centered hierarchy is forcing the sampler through a narrow neck, so aggressive integration keeps breaking at the same place. The non-centered panel stays much more isotropic.
Why it matters
This is the regime where reparameterization changes geometry more than NUTS path-length adaptation does.
Next move
Lower aggressiveness first, then compare how much more the non-centered parameterization helps than pure tuning does.
Two equivalent models, different geometry
What NUTS really changes
NUTS removes the need to hand-pick the leapfrog count. It does not repeal the neck of a centered funnel. If the geometry is wrong, path-length adaptation still drives a bad path through a bad region.