Chapter map

1. 1. Target-by-target modeling
2. 2. Intraview
3. 3. Juxtaview
4. 4. Paraview
5. 5. View generator
6. 6. Random-forest view model
7. 7. Cross-validation prediction
8. 8. Variance explained
9. 9. Late-fusion meta-model
10. 10. Normalized fusion coefficients
11. 11. Spatial gain
12. 12. Feature importance
13. 13. Interaction importance
14. 14. Across-sample aggregation
15. 15. Spatial nulls
16. 16. Governance and learning

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

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1. Target-by-target modeling

The central mathematical idea is concrete. One measured marker is predicted at a time.

A concrete situation

Protein T is the target across 500 cells.

What the mathematics does not license

A shared model can hide target-specific behavior. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Record every target and missing value. 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.

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2. Intraview

Here the geometry becomes operational. Other features at the same location predict the target.

A concrete situation

Markers A and B inside one cell predict T.

What the mathematics does not license

Co-expression is not intracellular regulation. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use it as the baseline. 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.

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3. Juxtaview

This step deserves a careful pause. Immediate neighbors are aggregated into local features.

A concrete situation

Touching cells contribute marker means.

What the mathematics does not license

Segmentation error changes adjacency. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Overlay every local edge. 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.

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4. Paraview

The useful question is not only how to calculate, but what the calculation assumes. A broader weighted neighborhood represents tissue context.

A concrete situation

Cells within 100 micrometres contribute.

What the mathematics does not license

Radius encodes a biological hypothesis. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Sweep justified ranges. 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.

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5. View generator

A small example makes the mechanism visible. A function maps coordinates and measurements into contextual predictors.

A concrete situation

Gaussian distance weights build a paraview.

What the mathematics does not license

Normalization changes dense regions. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Write the aggregation equation. 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.

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6. Random-forest view model

This is where modelling choices begin to affect evidence. Many decision trees predict the target from one view.

A concrete situation

Trees capture nonlinear thresholds and interactions.

What the mathematics does not license

Importance can favor high-variance predictors. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use permutation or grouped 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.

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7. Cross-validation prediction

The central mathematical idea is concrete. Held-out rows receive predictions from models not trained on them.

A concrete situation

Five folds produce one prediction per cell.

What the mathematics does not license

Random neighboring folds leak context. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Split by sample or region. 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.

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8. Variance explained

Here the geometry becomes operational. Prediction error is compared with target variance.

A concrete situation

R squared rises from 0.40 to 0.52.

What the mathematics does not license

Negative values are possible on holdout data. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report folds and uncertainty. 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.

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9. Late-fusion meta-model

This step deserves a careful pause. View predictions become inputs to a linear combiner.

A concrete situation

Intra, juxta and para predictions receive coefficients.

What the mathematics does not license

Correlated views share credit unstably. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Inspect coefficient covariance. 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.

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10. Normalized fusion coefficients

The useful question is not only how to calculate, but what the calculation assumes. Non-intercept coefficients are scaled to summarize relative contribution.

A concrete situation

Para receives 0.35 of normalized weight.

What the mathematics does not license

Relative contribution is not causal variance. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Compare ablations. 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.

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11. Spatial gain

A small example makes the mechanism visible. Multiview performance is compared with intraview baseline.

A concrete situation

Adding context improves held-out R squared by 0.08.

What the mathematics does not license

Gain may come from duplicated anatomy. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use coordinate permutations. 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.

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12. Feature importance

This is where modelling choices begin to affect evidence. Tree variance reduction ranks predictors within each view.

A concrete situation

Neighbor marker B helps predict T.

What the mathematics does not license

Importance does not give effect direction. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use response plots carefully. 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.

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13. Interaction importance

The central mathematical idea is concrete. Standardized feature importance is weighted by view contribution evidence.

A concrete situation

B-to-T is strong in the juxtaview.

What the mathematics does not license

The score is framework-specific. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Preserve raw components. 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.

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14. Across-sample aggregation

Here the geometry becomes operational. Interaction tables are combined after fitting individual samples.

A concrete situation

A median rank summarizes ten slides.

What the mathematics does not license

One large sample can dominate pooled fitting. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Show sample-level consistency. 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.

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15. Spatial nulls

This step deserves a careful pause. Coordinates or contextual features are permuted under a legitimate design.

A concrete situation

Within-slide shuffles remove neighborhood meaning.

What the mathematics does not license

Global shuffling can destroy batch structure. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Choose blocked nulls. 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.

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16. Governance and learning

The useful question is not only how to calculate, but what the calculation assumes. Version targets, views, radii, aggregators, forests, folds and fusion.

A concrete situation

A student compares three contextual predictors.

What the mathematics does not license

Importance is predictive evidence. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Write a MISTy view card. 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.

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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

MISTy asks where predictive information comes from. For each target marker, an intraview uses other markers at the same location, a juxtaview summarizes immediate neighbors and a paraview summarizes a broader radius. Additional views can represent pathways, cell types or anatomical regions. The radius, weighting and aggregation rule define the scientific meaning of a view, so two workflows using the same data can answer different spatial questions without either calculation being arithmetically wrong.

Each view receives its own predictive model; the published implementation uses random forests. Out-of-fold predictions from those view-specific models are combined by a linear late-fusion meta-model. The fusion coefficients estimate how much each view contributes to predicting a target beyond the intrinsic baseline. Comparing variance explained by the intraview alone with the multiview model offers a practical measure of added spatial information. It does not partition causal variance, because correlated views can share and redistribute predictive information.

MISTy then connects view contribution with feature importance. In a random forest, variance-reduction importance records how much a predictor helped split the training data. The framework standardizes these importances and weights them by evidence that the view contributes in the meta-model. A high interaction importance names a useful predictor-context combination, but it does not give a signed effect and does not prove regulation. Correlated markers, biased split opportunities and limited spatial coverage can change the ranking.

A strong validation plan permutes spatial positions within legitimate blocks, changes the juxta and para radii, compares alternative aggregators and holds out entire samples. Cross-validation folds must respect donors or sections; randomly splitting neighboring cells can leak nearly duplicated spatial context. Aggregate results only after confirming that individual samples are not dominated by one field of view. The happy payoff is a transparent question: which relationships help prediction inside a cell, beside it or across a wider tissue neighborhood?

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Frequently asked questions

What is MISTy?

MISTy is an explainable multiview framework that predicts each measured target from features representing intrinsic, immediate-neighborhood and broader spatial contexts.

What are the default views?

The intraview uses other markers at the same location, the juxtaview summarizes immediate neighbors and the paraview aggregates a wider radius.

Why use late fusion?

Separate view models make predictions first; a meta-model then estimates how much each context contributes without mixing all raw features into one opaque model.

What does variance explained measure?

It compares prediction error with total target variation. Improvement over the intraview indicates added predictive information from spatial views.

What is MISTy interaction importance?

It combines standardized view-model feature importance with evidence that the view contributes in the fusion model.

Does a large importance prove regulation?

No. It is predictive and unsigned. Correlation, shared anatomy, confounding and feature-importance bias can all produce a large value.

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Primary source and fact-check note

Jovan Tanevski, Aleix Gabor, Daniel Schapiro et al., “Explainable multiview framework for dissecting spatial relationships from highly multiplexed data,” Genome Biology 23, 97 (2022), DOI 10.1186/s13059-022-02663-5. 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

• Hotspot local gene relationships
• MERINGUE spatial cross-gene correlation
• SpatialDM ligand-receptor maps
• 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

View audit

Write the biological meaning of every view.

Radius audit

Compare juxta and para distance choices.

Split audit

Hold out donors, sections or regions.

Fusion audit

Inspect correlated view predictions and coefficients.

Importance audit

Preserve view contribution and raw feature importance.

Null audit

Permute spatial context within valid blocks.

Sample audit

Show individual-sample consistency before aggregation.

Final claim

Predictive importance is unsigned and non-causal.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Target-by-target modeling

Start with the chapter mechanism: One measured marker is predicted at a time. Recreate the concrete case: Protein T is the target across 500 cells. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: A shared model can hide target-specific behavior. Complete the transfer task: Record every target and missing value. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 2: Intraview

Start with the chapter mechanism: Other features at the same location predict the target. Recreate the concrete case: Markers A and B inside one cell predict T. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Co-expression is not intracellular regulation. Complete the transfer task: Use it as the baseline. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 3: Juxtaview

Start with the chapter mechanism: Immediate neighbors are aggregated into local features. Recreate the concrete case: Touching cells contribute marker means. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Segmentation error changes adjacency. Complete the transfer task: Overlay every local edge. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 4: Paraview

Start with the chapter mechanism: A broader weighted neighborhood represents tissue context. Recreate the concrete case: Cells within 100 micrometres contribute. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Radius encodes a biological hypothesis. Complete the transfer task: Sweep justified ranges. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 5: View generator

Start with the chapter mechanism: A function maps coordinates and measurements into contextual predictors. Recreate the concrete case: Gaussian distance weights build a paraview. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Normalization changes dense regions. Complete the transfer task: Write the aggregation equation. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 6: Random-forest view model

Start with the chapter mechanism: Many decision trees predict the target from one view. Recreate the concrete case: Trees capture nonlinear thresholds and interactions. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Importance can favor high-variance predictors. Complete the transfer task: Use permutation or grouped checks. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 7: Cross-validation prediction

Start with the chapter mechanism: Held-out rows receive predictions from models not trained on them. Recreate the concrete case: Five folds produce one prediction per cell. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Random neighboring folds leak context. Complete the transfer task: Split by sample or region. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 8: Variance explained

Start with the chapter mechanism: Prediction error is compared with target variance. Recreate the concrete case: R squared rises from 0.40 to 0.52. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Negative values are possible on holdout data. Complete the transfer task: Report folds and uncertainty. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 9: Late-fusion meta-model

Start with the chapter mechanism: View predictions become inputs to a linear combiner. Recreate the concrete case: Intra, juxta and para predictions receive coefficients. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Correlated views share credit unstably. Complete the transfer task: Inspect coefficient covariance. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 10: Normalized fusion coefficients

Start with the chapter mechanism: Non-intercept coefficients are scaled to summarize relative contribution. Recreate the concrete case: Para receives 0.35 of normalized weight. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Relative contribution is not causal variance. Complete the transfer task: Compare ablations. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 11: Spatial gain

Start with the chapter mechanism: Multiview performance is compared with intraview baseline. Recreate the concrete case: Adding context improves held-out R squared by 0.08. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Gain may come from duplicated anatomy. Complete the transfer task: Use coordinate permutations. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 12: Feature importance

Start with the chapter mechanism: Tree variance reduction ranks predictors within each view. Recreate the concrete case: Neighbor marker B helps predict T. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Importance does not give effect direction. Complete the transfer task: Use response plots carefully. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 13: Interaction importance

Start with the chapter mechanism: Standardized feature importance is weighted by view contribution evidence. Recreate the concrete case: B-to-T is strong in the juxtaview. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: The score is framework-specific. Complete the transfer task: Preserve raw components. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 14: Across-sample aggregation

Start with the chapter mechanism: Interaction tables are combined after fitting individual samples. Recreate the concrete case: A median rank summarizes ten slides. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: One large sample can dominate pooled fitting. Complete the transfer task: Show sample-level consistency. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 15: Spatial nulls

Start with the chapter mechanism: Coordinates or contextual features are permuted under a legitimate design. Recreate the concrete case: Within-slide shuffles remove neighborhood meaning. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Global shuffling can destroy batch structure. Complete the transfer task: Choose blocked nulls. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.

Workshop 16: Governance and learning

Start with the chapter mechanism: Version targets, views, radii, aggregators, forests, folds and fusion. Recreate the concrete case: A student compares three contextual predictors. Write target vectors, view matrices, neighbor weights, random-forest predictions, fold assignments, residual sums of squares, fusion coefficients and feature importances with names, dimensions and physical units where they exist. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge exactly one assumption named here: Importance is predictive evidence. Complete the transfer task: Write a MISTy view card. Compare the original result with the challenged result, explain which mathematical object changed, and mark measurement, model output, statistical evidence, biological interpretation and practical decision on separate lines. Reserve complete donors, sections or tissue regions outside tuning whenever the claim concerns transfer.