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The audit identifies the marginal accuracy distributions of two workflows, not the missing within-scene pairing. A copula supplies that dependence as an assumption. The Gold Standard is to climb an assumption ladder visibly: start with bounds, then show exactly how each dependence restriction narrows the scene-effect story.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the evidence
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE EVIDENCE
1. A copula is the missing pairing model
A copula links two marginal potential-outcome distributions into a joint distribution. For remote-sensing workflows, it describes whether scenes that would score highly under A also tend to score highly under B.
Randomisation does not identify this within-scene dependence because each scene reveals only one workflow outcome. Choosing a copula is therefore sensitivity analysis or structural modelling, not recovery of an observed pairing.
CHAPTER 2 OF 12 · DEFINE THE EVIDENCE
2. Rank invariance is a strong endpoint
Rank invariance pairs the same percentile under A and B. It says a scene’s latent rank is unchanged by the workflow, even though its score can move. This comonotone coupling often produces a narrow effect distribution.
Scenes can reorder when workflows respond differently to canopy, water, haze or urban texture. Use rank invariance as a named endpoint, not a neutral default. Rank similarity or bounded rank disruption can form weaker rungs.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A marginal scene ledger
Consider the same two sorted marginal accuracy distributions without row identities.
| Percentile position | Workflow A | Workflow B |
|---|---|---|
| Low | 70 | 68 |
| Lower-middle | 75 | 74 |
| Upper-middle | 80 | 81 |
| High | 85 | 83 |
| Mean | 77.5 | 76.5 |
Every compatible copula preserves these columns and the +1 mean effect, while the spread and harm fraction of scene-level effects can change sharply.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked assumption ladder
At the rank-invariant rung, pair equal positions: effects are +2, +1, −1 and +2, so 25% are negative. At the reverse-rank stress rung, effects are −13, −6, +6 and +17, so 50% are negative. Intermediate copulas can fill the space between these dependence patterns.
| Rung | Dependence statement | Illustrative effects | Negative fraction |
|---|---|---|---|
| Bounds only | Any compatible pairing | Set-valued | Bounded, not fixed |
| Rank invariance | Equal ranks pair | +2,+1,−1,+2 | 25% |
| Reverse-rank stress | Opposite ranks pair | −13,−6,+6,+17 | 50% |
| Intermediate copula | Parameter varied | Scenario-specific | Sensitivity curve |
The assumption ladder reports a curve or range, not one preferred percentage disguised as a measurement.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Parameterise dependence transparently
A Gaussian, Frank or other copula can index dependence with a parameter, but the parameter’s scientific meaning and tail behaviour differ. Translate parameter values into rank correlation, concordance or a concrete scene-ranking probability.
Do not treat an untestable parameter as if it were estimated from randomisation alone. Calibrate plausible ranges using repeated historical outcomes or panels only when those data support a defensible bridge to the counterfactual pairing.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Keep sampling uncertainty around every rung
Marginal distributions are estimated, so each copula scenario needs its own uncertainty calculation. A smooth sensitivity curve without confidence bands can look more certain than the design warrants.
Pointwise intervals answer one parameter value at a time. If the analyst searches many copula values and reports the most favourable one, simultaneous or worst-case uncertainty is needed for the claimed ladder.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: crop-stress classifiers
Two crop-stress workflows are randomised across fields. Rank invariance says a field that would rank in the 80th percentile under A also ranks there under B. A sensitivity ladder allows controlled rank disruption and asks how the fraction of harmed fields changes.
The transfer check keeps the field as the assignment unit, respects biome or crop blocks and separates dependence uncertainty from reference-label error. Pixel ranks cannot substitute for field-level potential outcomes.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Build decisions across the identified set
A robust decision can require acceptable loss under every copula in a declared sensitivity region. A less conservative rule can attach prior or policy weights, but those weights are additional inputs rather than design facts.
Choose the harm threshold and consequence function before viewing the ladder. If the recommendation changes under modest rank disruption, the honest result is dependence-sensitive and may justify a targeted paired or longitudinal audit.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Connect the heterogeneity branch
The Fréchet-bound guide supplies the assumption-light baseline. Randomisation-compatible heterogeneity bounds connect the set to scene-risk decisions.
Use robust decisions under accuracy sets when action must survive all credible dependence scenarios.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Limits and evidence checks
A copula is not guaranteed to transfer across biomes, sensors or reference-map versions. One global dependence parameter can hide different pairing structures by scene type. Stratification helps only where each block retains enough scenes.
Archive the marginals, copula family, parameter scale, calibration evidence, block definitions, harm threshold and uncertainty method. Show the no-assumption bounds beside every narrowed result.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Is rank invariance testable?
Not from a parallel randomised audit alone because both potential outcomes are never observed for one scene.
Which copula is best?
There is no universally best family. Choose a transparent sensitivity parameterisation tied to plausible scene concordance and compare families.
Can panel data help?
Yes, if past outcome dependence credibly informs the counterfactual pairing. That bridge remains an assumption and should be tested where possible.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Fan and Park develop sharp bounds for treatment-effect distributions. Frandsen studies partial identification with restrictions that tighten Fréchet bounds. Madigasekara and colleagues use copula-equality assumptions in panel data. The scene ledgers above are original teaching examples.
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