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A tracker can be uncertain in several different ways. Entropy describes spread, identity-switch mass describes a named error, and review should follow the consequences of that error.
Open the chapter contents · Remote Sensing root · Gold Standard of Education · How X Works
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 hypothesis spread
Route 2 · Work the identity ledger
Route 3 · Test marginals and truncation
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE HYPOTHESIS SPREAD
1. Treat the tracker output as a distribution over stories
A global tracker may retain several compatible explanations of the same detections. One hypothesis preserves two identities, another swaps them, and a third explains one detection as clutter or a missed object. Publishing only the highest-weight story hides how much probability remains elsewhere.
Hypothesis weights should be normalised over the retained, modelled set before they are interpreted. The resulting distribution can support entropy, marginal association probabilities and decision risk, but only relative to the hypotheses and likelihood model actually represented.
CHAPTER 2 OF 12 · DEFINE HYPOTHESIS SPREAD
2. Measure spread with hypothesis entropy
For weights p1 … pk, Shannon entropy is H = −Σ pi log2 pi. Entropy is zero when one retained hypothesis has all weight and increases as weight spreads. It is a summary of ambiguity, not a guarantee that the correct story was generated.
Entropy also depends on representation. Splitting one physical explanation into several nearly identical hypotheses can raise it. Compare entropy only after defining pruning, merging and labelling rules.
CHAPTER 3 OF 12 · WORK THE IDENTITY LEDGER
3. Work a three-hypothesis identity ledger
Two existing tracks, A and B, meet near two new detections, 1 and 2. The tracker retains H1 with weight 0.45: A→1 and B→2; H2 with weight 0.35: A→2 and B→1; and H3 with weight 0.20: A is missed while B→2. The entropy is about 1.51 bits.
The best story has only 0.45 weight, so an automatic choice of H1 carries 0.55 posterior mass on alternatives. The association A→1 has marginal probability 0.45; B→2 has marginal probability 0.65 because it occurs in H1 and H3.
| Hypothesis | Weight | Assignments | Identity implication |
|---|---|---|---|
| H1 | 0.45 | A→1, B→2 | identities continue |
| H2 | 0.35 | A→2, B→1 | pairwise switch |
| H3 | 0.20 | A missed, B→2 | one identity becomes dormant |
CHAPTER 4 OF 12 · WORK THE IDENTITY LEDGER
4. Turn identity-switch risk into a declared event
An identity switch is not merely “the second-best hypothesis exists.” Define the event relative to a reference: for example, the published continuation assigns A to the detection generated by B. Sum the posterior weights of retained hypotheses in which that event occurs.
For the ledger, choosing H1 makes the immediate pairwise-switch mass 0.35. The miss hypothesis creates a different future risk: A may later reconnect to the wrong object. Keep current switch, fragmentation and later relinking risks separate.
CHAPTER 5 OF 12 · WORK THE IDENTITY LEDGER
5. Use Bayes risk to decide whether to review
Suppose auto-accepting a wrong identity costs 20 units, correct acceptance costs zero, and expert review costs 3 units and reduces residual error probability to 0.05. For H1, auto-accept risk is 20 × 0.55 = 11. Review risk is 3 + 20 × 0.05 = 4, so review is preferred under these declared numbers.
| Action | Declared expected cost | What must be estimated |
|---|---|---|
| auto-accept H1 | 11.0 | probability and harm of a wrong global story |
| expert review | 4.0 | review cost and residual error after review |
| defer one revisit | context-dependent | delay harm and information gained |
CHAPTER 6 OF 12 · TEST MARGINALS AND TRUNCATION
6. Changed context: one story becomes dominant
Replace the weights with 0.75, 0.15 and 0.10. Entropy falls to about 1.05 bits, and the mass outside H1 is 0.25. Auto-accept risk becomes 5 units, still slightly above the illustrated review risk of 4.
If wrong-identity harm were eight rather than twenty, auto-accept risk would be 2 and review would not be justified. The same posterior supports different actions under different consequences.
CHAPTER 7 OF 12 · TEST MARGINALS AND TRUNCATION
7. Connect entropy to marginal association confidence
High hypothesis entropy can coexist with a stable marginal fact. Several global stories may differ elsewhere while all assign B→2. Conversely, low global entropy can hide one operationally critical link with weak marginal support if most uncertainty is concentrated on that link.
Publish both: a global spread measure and the marginals attached to identity-changing edges, births, deaths and misses. Continue through Near-Optimal Path Sets, Count Ambiguity and Marginal Association Confidence for the path-set view.
CHAPTER 8 OF 12 · TEST MARGINALS AND TRUNCATION
8. Account for truncation and model misspecification
Trackers prune low-weight hypotheses to control computation. Renormalising the survivors can make confidence look stronger than it was before pruning. Record retained mass, pruning thresholds and the number of distinct physical explanations removed or merged.
Posterior weights are conditional on detection probability, clutter, motion, birth and survival models. Calibration can fail when cloud, revisit geometry or target behaviour changes. Entropy cannot reveal a missing hypothesis.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Entropy is not the probability of error, and lower entropy is not always better. A confidently wrong model has low entropy. Identity-switch risk depends on the event definition and reference labelling. Marginal probabilities need not combine into one feasible global assignment when inspected independently.
Review is not automatically ground truth. Human analysts can share the same image ambiguity and may be biased by the proposed track. Measure reviewer disagreement and residual error on adjudicated cases.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Build an evidence check
Store the retained hypotheses before and after pruning, unnormalised and normalised weights, association graph, count distribution, marginal links, entropy, chosen action and review reason. Preserve sensor timing, masks, registration, appearance features and model version.
On labelled sequences, calibrate top-story probability and link marginals, then report identity switches, fragments, false births, false deaths and review yield. Stratify by entropy bands, top-two weight gap, revisit gap and crossing geometry. Compare auto-accept, defer and review policies at the same declared cost.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Questions readers should ask
Does 1.5 bits mean there are exactly three plausible stories?
No. Entropy summarises the weight distribution over the represented set; it is not a count and depends on pruning and representation.
Is the top hypothesis probability the confidence of every link inside it?
No. A link’s marginal probability is the sum of all retained hypotheses containing that link.
Should every high-entropy case be reviewed?
No. Review should be triggered by expected decision value, available alternatives and the consequences of being wrong.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue through verified routes
Begin with the Remote Sensing root, Gold Standard of Education and How X Works. Continue through Identity Confidence, Deferred Assignment and Retrospective Relinking, Tracklet Graphs, Global Flow and Nonlocal Identity Decisions and Birth–Death Priors, Fragmentation Penalties and Object-Count Selection.
Primary routes include Williams’ marginal multi-Bernoulli derivation of association-based trackers, Vo and Vo’s labeled random finite-set conjugate priors, and Kropfreiter and colleagues’ analysis of coalescence and repulsion in multitarget tracking. The three-hypothesis review ledger is an original teaching example.
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