The case
X₁, X₂ ~ N(μ, σ²), independent, σ known. History A reads (0, 10), history B (4, 6). Both means are 5 and both likelihoods for μ are proportional to exp(−(μ − 5)²/σ²), so the mean is sufficient and the histories are the same for μ.
- Source reappraisal. Instrument 1 is found miscalibrated and X₁ dropped. History A leaves 10, history B leaves 6. The mean cannot say which.
- Model revision. Investigators consider (1 − π)N(μ, σ²) + πN(μ + δ, σ²). Histories with the same mean and different spread carry different evidence for a second component, and the mean has discarded it.
What it can show
That sufficiency holds of a statistic under a model and a decision family, and lapses when either changes. A10’s ladder: a fixed model and family, where a statistic can suffice; new evidence about old sources; a new model or question; open-ended inquiry, where “which distinctions will matter is itself uncertain”.
What it cannot show
That the reasoning behind the estimate is what should have been kept. A10: “we do not need the researcher’s complete historical argument here. A source-indexed data record would suffice. This is an important objection to the idea that justificatory history specifically is what must be preserved.”