Chapter map

1. 1. Observed counts
2. 2. Expected counts
3. 3. Raw relative risk
4. 4. Poisson likelihood
5. 5. Log link
6. 6. Neighbour graph
7. 7. Structured effect
8. 8. Unstructured effect
9. 9. Precision parameters
10. 10. Partial pooling
11. 11. Posterior risk
12. 12. Exceedance probability
13. 13. BYM2 scaling
14. 14. Graph sensitivity
15. 15. Predictive checking
16. 16. Privacy and interpretation

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

---

1. Observed counts

A small example makes the mechanism visible. Fix Yi by region and period.

A concrete situation

Y=(8,2,5).

What the mathematics does not license

Small counts create noisy raw rates. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Preserve counts, not rounded rates. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

2. Expected counts

This is where modelling choices begin to affect evidence. Calculate Ei from population and risk strata.

A concrete situation

E=(5,5,5).

What the mathematics does not license

Crude expectations leave confounding. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Standardize relevant factors. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

3. Raw relative risk

The central mathematical idea is concrete. Compute Yi/Ei as a diagnostic.

A concrete situation

Raw SIRs are 1.6,.4,1.0.

What the mathematics does not license

A high SIR with few expected cases is unstable. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Add intervals. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

4. Poisson likelihood

Here the geometry becomes operational. Model Yi∼Poisson(Eiθi).

A concrete situation

The mean for area one is 5θ1.

What the mathematics does not license

Overdispersion may remain. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Check posterior predictive fit. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

5. Log link

This step deserves a careful pause. Write log θi=α+ui+vi.

A concrete situation

Each term adds on log risk.

What the mathematics does not license

Exponentiating changes interpretation. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report risk ratios. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

6. Neighbour graph

The useful question is not only how to calculate, but what the calculation assumes. Define which areas share influence.

A concrete situation

Areas 1–2 and 2–3 are adjacent.

What the mathematics does not license

Islands and bridges need explicit rules. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Publish the graph. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

7. Structured effect

A small example makes the mechanism visible. Use an intrinsic CAR prior for ui.

A concrete situation

u2 is centred near the mean of u1 and u3.

What the mathematics does not license

The prior is improper without constraints. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Apply sum-to-zero identification. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

8. Unstructured effect

This is where modelling choices begin to affect evidence. Give vi an independent Gaussian prior.

A concrete situation

Area one may have local noise beyond neighbours.

What the mathematics does not license

ui and vi can be confounded. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Inspect parameterization. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

9. Precision parameters

The central mathematical idea is concrete. Estimate or assign priors to spatial and iid variation.

A concrete situation

Higher precision means stronger shrinkage.

What the mathematics does not license

Vague priors can be unexpectedly informative. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Run prior predictive checks. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

10. Partial pooling

Here the geometry becomes operational. Combine likelihood and prior information.

A concrete situation

Low-information area two is pulled more strongly.

What the mathematics does not license

Smoothing is not data correction. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Preserve raw and smoothed views. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

11. Posterior risk

This step deserves a careful pause. Summarize θi from posterior samples or marginals.

A concrete situation

Area one has a risk distribution, not one truth.

What the mathematics does not license

Posterior mean alone hides uncertainty. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report intervals. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

12. Exceedance probability

The useful question is not only how to calculate, but what the calculation assumes. Calculate P(θi>1|data) when scientifically justified.

A concrete situation

A region has .82 probability above one.

What the mathematics does not license

Threshold maps invite multiple inspection. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

State the decision context. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

13. BYM2 scaling

A small example makes the mechanism visible. Separate total variance from the spatial mixing proportion.

A concrete situation

φ describes the structured share of variance.

What the mathematics does not license

Scaling depends on the graph. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Document software conventions. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

14. Graph sensitivity

This is where modelling choices begin to affect evidence. Compare plausible adjacency definitions.

A concrete situation

Ferry-connected islands alter neighbour sets.

What the mathematics does not license

Selecting the smoothest map is biased. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Predeclare alternatives. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

15. Predictive checking

The central mathematical idea is concrete. Simulate replicated counts from the fitted model.

A concrete situation

Replicates should match zeros and dispersion.

What the mathematics does not license

A neat map can fit poorly. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report discrepancies. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

16. Privacy and interpretation

Here the geometry becomes operational. Aggregate and communicate risk responsibly.

A concrete situation

Sparse areas are shown with wide intervals.

What the mathematics does not license

Maps can stigmatize communities. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Avoid causal labels and protect identities. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

---

A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

The Besag–York–Mollié model treats a disease map as a collection of noisy regional counts rather than a set of raw rates to be ranked. A common likelihood is Yi∼Poisson(Eiθi), where Yi is the observed count, Ei is the expected count after population and risk adjustment, and θi is relative risk. The log-risk is written as log θi=α+ui+vi, separating a spatially structured effect ui from an unstructured effect vi.

The structured component is usually an intrinsic conditional autoregressive field: each ui is encouraged to resemble the average of neighbouring spatial effects, with uncertainty depending on the number of neighbours and a precision parameter. The unstructured component captures area-specific variation that does not follow the neighbour graph. Together they partially pool unstable raw rates while allowing both smooth regional pattern and local noise.

Suppose three regions each have expected count 5 and observe 8, 2 and 5 cases. Their raw standardized incidence ratios are 1.6, 0.4 and 1.0. The BYM posterior does not simply replace these with an arithmetic average. It combines the Poisson evidence, the adjacency graph, priors and the balance between ui and vi, producing posterior risk distributions. A low-count region is usually shrunk more than a high-information region.

Smoothing improves estimation but can blur a genuine sharp boundary, import influence from a poor neighbour graph or hide denominator error behind a neat map. The original parameterization also makes structured and unstructured variance difficult to compare, motivating scaled BYM2 formulations. Responsible use reports posterior intervals and exceedance probabilities, tests graph and prior sensitivity, protects privacy and avoids reading causation into a smoothed association.

---

Frequently asked questions

What is the BYM likelihood?

A common disease-mapping form models Yi as Poisson with mean Eiθi, where Ei is expected burden and θi is relative risk.

What are ui and vi?

ui is a spatially structured effect linked through neighbours, while vi is independent area-specific variation.

What does spatial smoothing do?

It partially pools uncertain regional risks toward patterns supported by neighbours and the prior.

Why are expected counts important?

They adjust for population size and known risk composition before residual relative risk is mapped.

What should an uncertainty map show?

Posterior intervals or probabilities alongside posterior means, so low-information areas are not presented as certain.

Why consider BYM2?

A scaled reparameterization makes the total variance and structured-versus-unstructured mixing easier to interpret and assign priors to.

---

Primary source and fact-check note

Julian Besag, Jeremy York and Annie Mollié, “Bayesian image restoration, with two applications in spatial statistics,” Annals of the Institute of Statistical Mathematics 43, 1–20 (1991), DOI 10.1007/BF00116466. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• Moran’s I spatial autocorrelation
• Rogerson spatial chi-square
• How Mathematics Works

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: topic-specific validation and responsible use

Likelihood audit

Check counts, expected burden and Poisson assumptions.

Graph audit

Publish adjacency, islands and constraints.

Prior audit

Use prior predictive checks for variance and mixing.

Pooling audit

Compare raw and posterior risks with information levels.

Uncertainty audit

Map intervals or exceedance probabilities beside means.

Sensitivity audit

Test graph, prior and BYM versus BYM2 choices.

Predictive audit

Compare replicated and observed count features.

Final claim

Describe smoothed relative risk without converting association into cause.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Observed counts

Begin with this mechanism: Fix Yi by region and period. Recreate the concrete case: Y=(8,2,5). Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Small counts create noisy raw rates. Complete the transfer step: Preserve counts, not rounded rates. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 2: Expected counts

Begin with this mechanism: Calculate Ei from population and risk strata. Recreate the concrete case: E=(5,5,5). Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Crude expectations leave confounding. Complete the transfer step: Standardize relevant factors. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 3: Raw relative risk

Begin with this mechanism: Compute Yi/Ei as a diagnostic. Recreate the concrete case: Raw SIRs are 1.6,.4,1.0. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: A high SIR with few expected cases is unstable. Complete the transfer step: Add intervals. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 4: Poisson likelihood

Begin with this mechanism: Model Yi∼Poisson(Eiθi). Recreate the concrete case: The mean for area one is 5θ1. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Overdispersion may remain. Complete the transfer step: Check posterior predictive fit. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 5: Log link

Begin with this mechanism: Write log θi=α+ui+vi. Recreate the concrete case: Each term adds on log risk. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Exponentiating changes interpretation. Complete the transfer step: Report risk ratios. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 6: Neighbour graph

Begin with this mechanism: Define which areas share influence. Recreate the concrete case: Areas 1–2 and 2–3 are adjacent. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Islands and bridges need explicit rules. Complete the transfer step: Publish the graph. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 7: Structured effect

Begin with this mechanism: Use an intrinsic CAR prior for ui. Recreate the concrete case: u2 is centred near the mean of u1 and u3. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: The prior is improper without constraints. Complete the transfer step: Apply sum-to-zero identification. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 8: Unstructured effect

Begin with this mechanism: Give vi an independent Gaussian prior. Recreate the concrete case: Area one may have local noise beyond neighbours. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: ui and vi can be confounded. Complete the transfer step: Inspect parameterization. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 9: Precision parameters

Begin with this mechanism: Estimate or assign priors to spatial and iid variation. Recreate the concrete case: Higher precision means stronger shrinkage. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Vague priors can be unexpectedly informative. Complete the transfer step: Run prior predictive checks. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 10: Partial pooling

Begin with this mechanism: Combine likelihood and prior information. Recreate the concrete case: Low-information area two is pulled more strongly. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Smoothing is not data correction. Complete the transfer step: Preserve raw and smoothed views. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 11: Posterior risk

Begin with this mechanism: Summarize θi from posterior samples or marginals. Recreate the concrete case: Area one has a risk distribution, not one truth. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Posterior mean alone hides uncertainty. Complete the transfer step: Report intervals. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 12: Exceedance probability

Begin with this mechanism: Calculate P(θi>1|data) when scientifically justified. Recreate the concrete case: A region has .82 probability above one. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Threshold maps invite multiple inspection. Complete the transfer step: State the decision context. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 13: BYM2 scaling

Begin with this mechanism: Separate total variance from the spatial mixing proportion. Recreate the concrete case: φ describes the structured share of variance. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Scaling depends on the graph. Complete the transfer step: Document software conventions. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 14: Graph sensitivity

Begin with this mechanism: Compare plausible adjacency definitions. Recreate the concrete case: Ferry-connected islands alter neighbour sets. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Selecting the smoothest map is biased. Complete the transfer step: Predeclare alternatives. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 15: Predictive checking

Begin with this mechanism: Simulate replicated counts from the fitted model. Recreate the concrete case: Replicates should match zeros and dispersion. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: A neat map can fit poorly. Complete the transfer step: Report discrepancies. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 16: Privacy and interpretation

Begin with this mechanism: Aggregate and communicate risk responsibly. Recreate the concrete case: Sparse areas are shown with wide intervals. Write observed counts, expected counts, raw risk ratios, log risks, neighbour edges, structured effects, unstructured effects, precisions, mixing proportions, posterior means, intervals, exceedance probabilities and predictive counts with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Maps can stigmatize communities. Complete the transfer step: Avoid causal labels and protect identities. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.