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Mixture labels can swap, classes can split and two posterior draws can disagree while making similar predictions. A posterior co-clustering matrix asks the label-free question that matters: how often do two protocols belong together? A consensus map then offers one readable summary without erasing the uncertainty around it.
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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 decision
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 DECISION
1. A partition is the object, not its label names
In one posterior draw, ordinary protocols may be class 1; in another, the same group may be class 3. Label switching changes names but not the underlying partition.
Mechanism summaries should therefore start from label-invariant quantities. Co-clustering probabilities, predictive contrasts and partition losses remain meaningful when component labels exchange.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Build the posterior similarity matrix
For protocols i and j, the posterior similarity entry is the fraction of retained draws in which they occupy the same cluster. Values near one support stable grouping; values near zero support separation; middle values expose unresolved boundaries.
The matrix is not itself a hard clustering. Pairwise probabilities need not describe one perfectly transitive partition, so a consensus rule and loss function are still required.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A co-clustering ledger
Consider four polymer validation protocols after integrating over uncertain class count and labels.
| Pair | Co-clustering probability | Reading | Mechanism caution |
|---|---|---|---|
| A–B | 0.93 | Stable pair | Shared mechanism remains a hypothesis |
| A–C | 0.48 | Boundary pair | Partition is unresolved |
| B–C | 0.41 | Mostly separate | Sensitive to component family |
| C–D | 0.82 | Likely pair | Check temperature confounding |
Order the heat map by the consensus partition, but retain numerical probabilities and protocol metadata. A polished colour block must not turn 0.52 into certainty.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked consensus-partition comparison
Three candidate summaries are scored under a declared posterior partition loss. Lower expected loss is better.
| Consensus map | Groups | Expected loss | Held-out mechanism check |
|---|---|---|---|
| One class | ABCD | 3.8 | Misses low-contrast subgroup |
| Two classes | AB | CD | 1.6 | Passes three of four checks |
| Three classes | AB | C | D | 1.9 | D class unstable |
A consensus choice compresses the posterior; it does not become the posterior. Publish credible alternatives or a partition uncertainty ball when decisions change across nearby summaries.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Stability needs more than one chain
Compare posterior similarity matrices across chains, initialisations, component families and prior settings. Large matrix differences can reveal poor mixing or a clustering story driven by one modelling choice.
Use held-out protocols and replicate measurements. A partition can be internally stable yet predict new mechanism-sensitive evidence badly.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Turn consensus groups into mechanism maps carefully
Attach spectroscopy, thermal transitions, morphology and process conditions after the partition summary is stable enough to inspect. Mechanism names need external evidence; they are not generated automatically by clustering.
Show uncertainty at the protocol boundary. If a protocol co-clusters with two groups at similar probabilities, call it ambiguous or transitional instead of forcing a clean pathway label.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: ageing trajectories
Transfer the workflow to polymer-ageing trajectories. Two specimens may share an early degradation pattern but diverge after humidity exposure, producing time-window-dependent co-clustering.
The transfer check specifies trajectory distance, censoring, replicate structure and the prediction target. It tests whether the consensus map survives a later observation window.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the mixture branch
The unknown-class-count guide explains split–merge uncertainty. The mixture-hierarchy guide develops class-specific pooling.
Use the hierarchy-sensitivity guide when consensus changes after information is pooled differently.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
A high co-clustering probability does not prove a common causal pathway. Shared nuisance conditions can also produce a stable statistical group.
Consensus clustering is not majority voting over class labels. It is a decision summary under a stated loss, and different losses can select different maps.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive posterior draws, class-count support, similarity matrices, chain comparisons, partition loss, candidate consensus maps, uncertainty summaries, held-out protocols and mechanism assays.
Report unstable pairs, smallest groups, prior sensitivity and decisions under nearby partitions. Preserve protocol identifiers so scientific reviewers can return from the heat map to raw evidence.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Why not choose the most frequent sampled partition?
The partition space is enormous, so one exact partition may appear rarely even when its pairwise structure is well supported.
Can the matrix determine the number of groups?
Only through an added decision rule or clustering criterion. The matrix alone contains pairwise posterior probabilities.
What if two consensus rules disagree?
Show both, compare posterior loss and test whether the scientific decision changes. Disagreement is useful uncertainty, not a formatting problem.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Fritsch and Ickstadt develop criteria based on the posterior similarity matrix. Wade and Ghahramani study Bayesian partition point estimates and credible balls. Dahl, Johnson and Müller examine search algorithms and loss functions for Bayesian clustering. The polymer ledgers above are original teaching examples.
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