A noise-free curve yields a zero-width confidence interval, which is correct arithmetic and a misleading claim

[marcohost33-maker]

Found in round 20 of PR #121, recorded rather than fixed because the honest fix is a decision about
what an interval is allowed to mean, not a patch.

What happens

Feed fitting/bootstrap.py a curve that the model describes exactly, with no noise: the estimated
residual scale is about 1e-15, every parametric resample is essentially the same curve, and the
returned BCa interval has width 0.0.

Why this is not simply a bug

Inside the parametric bootstrap this is arithmetically correct: the resampling distribution really
does have no spread. The defect is at the interface. A confidence interval of width zero is read by
every downstream consumer as maximal certainty, when what actually happened is that the
uncertainty model had nothing to work with.

That makes it the same family as the findings this PR closed — the number is not wrong, the
claim carried by the number is.

Options, none of them free

1. Refuse to emit a finite interval below a residual-scale floor and mark uncertainty unavailable.
   Honest, but introduces a threshold that must itself be justified.
2. Emit the interval and a companion flag saying the resampling distribution was degenerate.
   Cheapest, but relies on consumers reading the flag.
3. Leave as is and document. Weakest: a documented limit prevents nothing.

Maintainer decision — this changes what a published interval means.

Discrimination requirement whichever route is taken

The chosen behaviour must fire on the noise-free curve and stay silent on a curve with ordinary
noise. Both directions in one test; the silent direction is the one that is usually forgotten.

[marcohost33-maker]

Triage (read-only, 2026-09-07): still open on main; the maintainer decision this issue asks for is largely pre-settled by an external reference implementation, and the exact line that produces the misleading claim is identifiable.

1. State on the default branch

src/liouscope/fitting/bca_ci on main carries no degeneracy handling of any kind — no flag, no NaN, no warning. PR #143 (fix(#125): fail closed on degenerate BCa intervals) implements exactly the behaviour discussed here, but it is OPEN with baseRefName = chatgpt/issue-124-residual-scaling, i.e. behind a three-deep open stack:

#143 (issue-125-degenerate-bca)  -> base chatgpt/issue-124-residual-scaling
#134 (issue-124-residual-scaling) -> base pr107-fix        [OPEN]
#121 (pr107-fix)                  -> base main             [OPEN]

So even a merge of #143 changes nothing on main. The issue is correctly open.

2. Root cause: the tie half-correction erases the degeneracy before it can be detected

Reproduced standalone (the bca_ci arithmetic copied verbatim from main, degenerate resample, B=1000):

A degenerate      ci = [1.234567 1.234567]   width = 0.0
   raw_prop = 0.500000  z0 = 0.000000  a = 0.000  q_lo = 0.025  q_hi = 0.975
   any NaN? False        any warning? None
B ordinary noise  ci = [1.14840517 1.33460811]  width = 0.186
C scipy BCa, same shape of input:
   ConfidenceInterval(low=nan, high=nan)   warnings: [RuntimeWarning, DegenerateDataWarning]

The relevant lines are bootstrap.py:109-112 on main:

prop_below = float(
    np.mean(boot_j < theta_hat[j]) + 0.5 * np.mean(boot_j == theta_hat[j])
)
prop_below = float(np.clip(prop_below, 1.0e-9, 1.0 - 1.0e-9))
z0 = ndtri(prop_below)

When every replicate equals theta_hat, the strict-< term is 0.0 and the tie term is 0.5 * 1.0, so prop_below is exactly 0.5 and z0 is exactly 0 — a perfectly ordinary, unbiased-looking bias correction. Textbook BCa without the tie term would give prop_below = 0 -> z0 = -inf -> NaN, which is precisely the signal SciPy emits. The S3 half-correction (added 2026-06-04 for a legitimate reason — bounded/clipped parameters) therefore does not merely soften the degenerate case, it converts the only available degeneracy signal into a confident zero. The np.clip(..., 1e-9, 1-1e-9) on the next line is a second, independent NaN-suppressor that would clamp z0 even without the tie term.

Upstream of this, gls.py:116 floors the scale at sqrt(1e-30) = 1e-15, which is exactly the "residual scale about 1e-15" reported above — the floor is where the 1e-15 comes from, it is not an accident of the data.

3. The threshold objection against Option 1 does not hold

The issue rejects Option 1 partly because "it introduces a threshold that must itself be justified". SciPy's scipy.stats.bootstrap(method='BCa') withholds the interval without any noise floor: it detects degeneracy structurally (bootstrap distribution all-identical / z0 non-finite), returns NaN endpoints and raises DegenerateDataWarning, with the documented advice to "consider using another method or inspecting data for indications that other analysis may be more appropriate (e.g. all observations are identical)" — documented unchanged from SciPy 1.10 (2023) through the current 1.18 manual (retrieved 2026-09-07). Efron's own construction supports this reading: BCa is defined through a finite z0 and a; on a degenerate resample neither exists, so there is no BCa interval to report, rather than a zero-width one (Efron 1987; DiCiccio & Efron, Bootstrap Confidence Intervals, Statistical Science 11(3), 1996).

That collapses the decision space: a structural fail-closed rule is Option 1 minus the threshold that made Option 1 expensive, and it is what the most widely used reference implementation already does. PR #143's description matches this (per-parameter NaN + RuntimeWarning, no invented floor) — the semantic decision requested here therefore has an external anchor, not just a preference.

4. Note on the discrimination requirement

The silent direction holds in the repro (width = 0.186 under ordinary noise), but the guard as described must be tested against prop_below == 0.5-by-ties as well as by symmetry: a genuinely symmetric, well-sampled distribution also yields z0 = 0. z0 alone cannot separate the two cases; the discriminator has to be the spread of boot_j (or its all-identical-ness), not the bias correction. A test that only asserts on z0 would pass while remaining fail-open.

Sources: scipy.stats.bootstrap manual · DiCiccio & Efron, Bootstrap Confidence Intervals, Statist. Sci. 11(3) 1996

No source file was modified for this triage; the reproduction was built outside the repository.