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When every reference source can be wrong, a latent-class model can organise the uncertainty, but it cannot identify truth without enough evidence and defensible assumptions.
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Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Define the latent model
Route 2 · Work the agreement ledger
Route 3 · Test dependence and transfer
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE LATENT MODEL
1. Accuracy when no label is unquestioned
A classified map, an interpreter and an existing product can all be wrong. Latent-class models treat the unobserved land-cover class as a hidden variable and estimate how each observable label source responds to it. This can be useful when field reference is absent or imperfect, but the result is model-based evidence, not recovered ground truth.
The hard part is identifiability. Several combinations of prevalence and source error rates can produce the same observed agreement table. Extra label sources, populations, constraints or prior information help only when their assumptions are explicit and defensible.
CHAPTER 2 OF 12 · DEFINE THE LATENT MODEL
2. The binary latent-class model
Let Z be an unobserved binary class such as forest or non-forest. For each source j, sensitivity is Pr(source j says forest | Z=forest), while specificity is Pr(source j says non-forest | Z=non-forest). Prevalence π is Pr(Z=forest).
Under conditional independence, the probability of a pattern of source labels is a mixture of two products: one product within the forest class and one within the non-forest class. That factorisation supplies the estimating equations. If sources remain dependent after conditioning on Z, the equations are wrong unless dependence terms are added.
CHAPTER 3 OF 12 · WORK THE AGREEMENT LEDGER
3. An observed agreement ledger
Suppose a map M and reviewer R label 100 sampled units.
| Map M | Reviewer R forest | Reviewer R non-forest | Row total |
|---|---|---|---|
| Forest | 40 | 15 | 55 |
| Non-forest | 10 | 35 | 45 |
The table has four cell probabilities that sum to one, so it supplies three independent quantities. A two-source, one-population latent-class model has five basic unknowns: prevalence plus sensitivity and specificity for each source. The data therefore cannot identify all five without additional information. A solver may still return numbers, but numerical convergence is not identification.
CHAPTER 4 OF 12 · WORK THE AGREEMENT LEDGER
4. Why several truths fit the same table
High agreement can arise because both sources are accurate, because one class dominates, or because both sources make correlated errors. Disagreement does not reveal which source is wrong. Even under conditional independence, different parameter combinations may fit when there are too few degrees of freedom.
Profile likelihoods, posterior intervals and multiple starting values can expose flat directions. If fitted accuracy changes sharply when one plausible prior or constraint moves, the conclusion should be reported as assumption-sensitive.
CHAPTER 5 OF 12 · WORK THE AGREEMENT LEDGER
5. Adding a second population
The Hui–Walter construction applies two tests to two populations with different prevalences, while assuming the tests have the same sensitivities and specificities in both populations and are conditionally independent. Two 2×2 tables provide six independent probabilities; the model has four test-performance parameters and two prevalences.
| Added evidence | New information | New assumption |
|---|---|---|
| Second region | Different class mixture | Error rates transport across regions |
| Third label source | More response patterns | Dependence structure is specified |
| Small audited subset | Anchors selected error rates | Audit is representative |
| Informative prior | Regularises weak directions | Prior knowledge is credible |
Different prevalence alone is insufficient if the map behaves differently across biomes, seasons or sensor conditions. Test the transportability claim using covariates or stratified fits rather than assuming it silently.
CHAPTER 6 OF 12 · TEST DEPENDENCE AND TRANSFER
6. Conditional dependence from shared imagery
Map M and reviewer R may both rely on the same optical image. Cloud shadow, mixed pixels and temporal mismatch can then make them err together even after the true class is fixed. Conditional independence interprets this excess agreement as stronger accuracy than the evidence warrants.
Possible responses include a dependence parameter, random effects, a third source with genuinely different evidence, or a targeted field audit. Dependence models add parameters and can reopen identifiability problems. Report sensitivity across plausible structures rather than selecting the model with the most flattering accuracies.
CHAPTER 7 OF 12 · TEST DEPENDENCE AND TRANSFER
7. Changed-context transfer: shared training labels
Suppose a second land-cover product was trained on labels created by the same interpreter who supplies reference labels. Treating product and interpreter as independent sources double-counts shared information. Their agreement can be almost guaranteed in familiar locations while both fail on a new landscape.
The transfer check is to map the evidence lineage. Group sources by shared imagery, training data and adjudication. Refit after removing one source from each shared lineage, then compare estimates on a geographically separated audit. This continues the work on probabilistic reference labels and response matrices by moving from known response uncertainty to an unidentified latent truth.
CHAPTER 8 OF 12 · TEST DEPENDENCE AND TRANSFER
8. Label switching and class meaning
A latent model can exchange the names of its two classes and keep the same likelihood. In a binary case, constraints such as sensitivity plus specificity exceeding one may orient sources toward the intended labels, but such constraints are substantive assumptions.
For multiple classes, matching components to forest types can be harder. Use external semantic anchors, stable class definitions and posterior predictive checks. A component is not “mangrove” merely because it has the desired map pattern.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Sampling design still matters
Latent modelling does not cancel unequal inclusion probabilities. If samples are stratified by map class or region, likelihood or estimating procedures must respect the design where required for the target quantity. Site-specific accuracy and area-wide prevalence are different estimands.
Keep the sample frame, inclusion probabilities, nonresponse and spatial support. A pixel label, a polygon majority and a field plot may refer to different units even when their class names match.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Limits and practical evidence checks
Latent classes are useful only relative to a declared model. Check identifiability analytically where possible, not only through a Hessian. Use simulated data generated from fitted parameters to see whether observed agreement patterns, region differences and marginal class rates are reproduced.
Fit plausible dependence alternatives, inspect multiple modes, disclose priors and constraints, and reserve independent audit data. Report accuracy intervals and model sensitivity together. If the evidence cannot distinguish competing models, say so; “unidentified” is a result, not a software failure.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does a latent class become the true class?
No. It is an unobserved model component linked to observed labels through assumptions. Physical interpretation requires external evidence.
Are three sources always enough?
No. Their dependence, class count, sampling design and parameter constraints determine identification. Three nearly identical sources may add little independent information.
Can an imperfect audited subset help?
Yes, if its error process is modelled and its selection is understood. Calling it perfect when it is not can bias the entire system.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Hui and Walter introduced a two-test, two-population approach in Estimating the Error Rates of Diagnostic Tests. Foody examines imperfect reference data and latent-class assessment in Remote Sensing of Environment. These motivate the framework; the map-reviewer ledger and lineage checks above are original teaching examples.
Continue through the Remote Sensing root, probabilistic reference labels and reviewer disagreement and adjudication. The Gold Standard of Education and How X Works connect identification to transferable reasoning.
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