Mean/median prediction-centering for every noise model (complete the location axis; neg_bin median)

[wshlavacek]

Summary

The location axis (ADR-0011 / ADR-0024) lets the model prediction be interpreted as a chosen summary of the noise distribution -- mean or median. PyBNF should offer both for every noise model / likelihood, with median the default. (Median aligns with PEtab v2's convention, but this is a PyBNF capability decision -- PyBNF deliberately supports more than the PEtab v2 subset.)

Current state

• Gaussian (normal / lognormal): both supported. On a linear scale mean = median (coincide); on a log scale median is the default (offset 0) and mean adds the Gaussian moment correction mu = log(pred) - ln(base)*sigma**2/2 (noise/scale.py::mean_offset).
• Per-observable selection via the location = mean|median field on the noise_model line (ADR-0024). A global noise_location default is planned.

Gaps to close

1. neg_bin median. neg_bin is parameterized directly by its mean (the prediction is the mean), so location = mean is the current (only) interpretation; location = median raises "not implemented" today (objective._apply_location). The negative-binomial median has no closed form, so centering the prediction on it needs a numeric CDF inversion (place prediction at the median by solving nbinom.cdf(median; r, p) = 0.5). Implement so location = median works for neg_bin.

2. log-laplace mean. Laplace's moment correction on a log scale differs from Gaussian's: the mean of a (natural-log) log-Laplace is e^m / (1 - b**2) (for |b| < 1), not e^(m + sigma**2/2). scale.mean_offset currently implements only the Gaussian correction (noted in ADR-0011 / ADR-0024), so location = mean on a log-scale Laplace would be numerically wrong today. Give the Laplace family its own moment correction (make the offset family-aware).

Default

Median across the board (the ADR-0024 default). neg_bin's existing default is mean; once median is implemented, decide whether to flip neg_bin's default to median (a behavior change) or keep mean as its default with median opt-in (backward-compat consideration).

Acceptance

Every (noise family x additive scale) supports both mean and median centering, correctly, validated against scipy oracles (nbinom, laplace, lognorm). Relates to ADR-0011, ADR-0021, ADR-0024.

[wshlavacek]

Paused pending #424. The mean-vs-median convention/default question raised here (the "decide whether to flip neg_bin's default" footnote) turns out to be a symptom of a broader unsettled policy: the per-family defaults are inconsistent (Gaussian→mean, Laplace→median, lognormal→median, neg_bin→mean) and PEtab v2 (median) vs backward-compat (mean) are in tension. Split out as #424 (the policy: go-forward convention + backward-compat mechanism + possible .conf era marker). This issue stays the capability (implement both centerings correctly for every family — neg_bin median via CDF inversion, log-laplace mean via a family-aware moment correction), to be wired up under whatever convention #424 settles.