Curvature, resampling, walking — run all three and read the disagreements.
This site has now built three independent ways of pricing uncertainty: the curvature route (Wald intervals from the Hessian), the resampling route (bootstrap percentile intervals from the loop), and the sampling route (credible intervals from the walked posterior). They rest on different assumptions, so their pattern of agreement is itself information. Three scenarios, computed live from this site's validated datasets and seeds:
| Route | 95% interval | How it got there |
|---|---|---|
| computing… | ||
All three agree (the well-behaved case): every route is trustworthy here; report the parametric one, which readers know, and move on. The agreement is not redundancy — it is the evidence that the assumptions behind the fast method hold.
Bootstrap and posterior agree, Wald differs (the small-skewed case):
the two methods that follow the surface's actual shape are outvoting the one that
assumed a quadratic. Believe the shape-followers, and be suspicious of every
symmetric $\pm 1.96\,\text{SE}$ statement made about this parameter — including
the ones inside summary().
Every route strains (the boundary case): Wald leaks into impossible territory; the bootstrap respects the boundary but goes granular (only 21 possible resample proportions exist, and its interval snaps to that lattice, here bottoming out at exactly zero); the posterior stays smooth and in-bounds. Near boundaries with tiny $n$, the sampling route is the only comfortable one — which is precisely why the Bayesian page opened with this example.
Practical habit. The cheap routes run in milliseconds — so run them both (Wald and bootstrap) as a matter of routine, and escalate to the walked posterior when they split or when the answer matters enough to pay for. The point of having three traditions is not choosing a team; it is triangulation.