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A two-class mixture may look convincing because it divides ordinary and low-contrast validation conditions. Yet the data may also support one broad class or three narrower ones. The useful question is not only “How many peaks?” but which decisions and mechanism checks remain stable when classes split, merge or exchange labels.
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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. The class count is part of the model
Fixing K=2 treats the number of latent validation mechanisms as known. Allowing K to vary places uncertainty on the partition as well as on component means, spreads and mixing weights.
Posterior support for K is conditional on the component family, priors and measurement model. Slight misspecification can create extra components that approximate skewness or heavy tails rather than reveal new chemistry.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Split and merge moves test neighbouring explanations
A split proposes dividing one component into two; a merge proposes combining two components. Reversible-jump, mixture-of-finite-mixtures and overfitted-mixture samplers use different machinery, but all must explore competing partitions rather than remain trapped in one labelling.
Poor movement between K values can mimic certainty. Inspect transition counts, repeated chains and class-invariant predictive summaries before interpreting a dominant class count.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A class-count ledger
Suppose eight protocols are fitted with one, two and three validation classes.
| Model | Posterior support | Held-out log score | Smallest class | Mechanism interpretation |
|---|---|---|---|---|
| K=1 | 0.18 | −42.6 | All 8 | One broad population |
| K=2 | 0.57 | −38.9 | 3 protocols | Ordinary vs low contrast |
| K=3 | 0.25 | −38.6 | 1 protocol | Adds a tiny high-temperature class |
The 0.57 support for K=2 is not proof of exactly two mechanisms. It is evidence within the declared candidate models.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked split–merge decision sensitivity
A release threshold requires predicted trigger sensitivity above 70%. Average over class-count uncertainty rather than selecting K first.
| Model | Protocol F prediction | Release decision | Posterior weight |
|---|---|---|---|
| K=1 | 72% | Release | 0.18 |
| K=2 | 68% | Hold | 0.57 |
| K=3 | 74% | Release | 0.25 |
| Model-averaged | 70.1% | Borderline | 1.00 |
Because the decision sits on the threshold, collect mechanism-specific evidence or use a conservative rule rather than hiding the split–merge instability.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Check mechanisms through predictive discrepancies
Choose discrepancies tied to material science: false-trigger counts by temperature, residual optical contrast, transition timing, crystallinity strata or spectroscopy-confirmed pathway labels.
A model can reproduce the marginal sensitivity histogram while failing the joint pattern that distinguishes mechanisms. Posterior predictive checks should target that failure, not only a total count.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Separate clusters from causal pathways
Latent classes summarise distributional heterogeneity. They become mechanism claims only when external assays, experimental interventions or stable covariate relationships support that interpretation.
Report class-invariant predictions and uncertainty over partitions. Relabelling for presentation must not make uncertain membership appear observed.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: blend morphology
Transfer the workflow to polymer-blend morphology where microscopy features may suggest dispersed, co-continuous and interfacial regimes. Image segmentation uncertainty can create apparent extra classes.
The transfer check uses held-out specimens, replicate imaging and independent thermal or mechanical measurements. A one-specimen class is a prompt for investigation, not a discovered phase.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the mixture branch
The mixture-hierarchy guide develops class-specific pooling. The hierarchy-sensitivity guide tests whether information travels along the right experimental structure.
Use the sparse-validation guide when the one-population model remains adequate.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
A larger posterior probability for K does not mean each component is well separated or scientifically meaningful. Near-empty and overlapping components can make allocation probabilities unstable.
Information criteria, reversible-jump sampling and overfitted mixtures answer related but different questions. Agreement across methods is reassuring but not a substitute for predictive and mechanism checks.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive candidate component families, prior on K, weight priors, split–merge proposals, initialisations, chain transitions, allocation uncertainty, held-out units and mechanism-sensitive discrepancies.
Report support for K, predictive scores, smallest classes, class-invariant decisions and sensitivity to skewed or heavy-tailed single-component alternatives. Retain raw protocol-level evidence.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does more data always reveal the true K?
Not under model misspecification. Extra components may continue to approximate a wrong component family.
Should a tiny class be deleted?
Not automatically. Check provenance and mechanism evidence, then show decisions with and without it.
Can I average across K?
Yes, for predictions and decisions, provided the model probabilities and within-model calculations are trustworthy and label-invariant.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Richardson and Green develop Bayesian mixture analysis with an unknown number of components. Miller and Harrison study mixtures of finite mixtures with a prior on component count. Cai, Campbell and Broderick show why finite mixtures may not reliably learn component count under misspecification. The polymer ledgers above are original teaching examples.
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