Chapter map

1. 1. Expression feature vectors
2. 2. Cell-type labels
3. 3. Multivariate normal layer
4. 4. Shared covariance
5. 5. Spatial-domain labels
6. 6. Domain composition vector
7. 7. Categorical bridge
8. 8. Neighbourhood graph
9. 9. Potts interaction
10. 10. Joint posterior
11. 11. Bayesian priors
12. 12. Efficient inference
13. 13. Multi-section sharing
14. 14. Cell-type proportions
15. 15. Scale-specific outputs
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. Expression feature vectors

This is where modelling choices begin to affect evidence. Each cell has normalized molecular features selected for modeling.

A concrete situation

A cell is represented by ten spatially variable components.

What the mathematics does not license

Feature selection can omit rare programs. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Repeat robust gene sets. 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. Cell-type labels

The central mathematical idea is concrete. A hidden categorical variable assigns each cell to one type.

A concrete situation

A neuron-like cell is provisionally type two.

What the mathematics does not license

A discrete type may hide continua. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Inspect posterior probabilities. 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. Multivariate normal layer

Here the geometry becomes operational. Expression depends on a cell-type-specific mean and shared covariance.

A concrete situation

Type two has a distinct two-feature centre.

What the mathematics does not license

Gaussian tails and shared covariance may misfit. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Plot residuals by type. 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. Shared covariance

This step deserves a careful pause. One matrix describes within-type variation across all types.

A concrete situation

Correlated genes form tilted probability ellipses.

What the mathematics does not license

Rare types may need different covariance. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Compare flexible 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.

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5. Spatial-domain labels

The useful question is not only how to calculate, but what the calculation assumes. A second hidden variable assigns each cell to a regional domain.

A concrete situation

Layer four is domain three.

What the mathematics does not license

Domain count shapes segmentation. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Report several counts. 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. Domain composition vector

A small example makes the mechanism visible. Each domain contains categorical probabilities over cell types.

A concrete situation

Domain three is 70 per cent neurons and 30 per cent glia.

What the mathematics does not license

Estimated composition depends on type labels. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Propagate 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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7. Categorical bridge

This is where modelling choices begin to affect evidence. Cell type is drawn conditional on the cell’s domain.

A concrete situation

A domain rich in type two increases its prior probability.

What the mathematics does not license

Unusual cells can be overruled by context. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Inspect expression-versus-context conflict. 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. Neighbourhood graph

The central mathematical idea is concrete. Coordinates define edges within every tissue section.

A concrete situation

A kNN graph links local cells.

What the mathematics does not license

Density changes physical scale. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Map edge lengths. 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. Potts interaction

Here the geometry becomes operational. Agreeing neighbouring domain labels increase prior support.

A concrete situation

A smooth layer beats a checkerboard partition.

What the mathematics does not license

Strong beta erases thin structures. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Publish beta sensitivity. 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. Joint posterior

This step deserves a careful pause. Expression, type, domain composition and graph smoothness combine.

A concrete situation

One labeling balances molecular fit and spatial coherence.

What the mathematics does not license

Posterior modes can be multimodal. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use multiple chains or starts. 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. Bayesian priors

The useful question is not only how to calculate, but what the calculation assumes. Parameters receive distributions that regularize sparse evidence.

A concrete situation

Rare-type proportions shrink toward a prior.

What the mathematics does not license

Priors can dominate small domains. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Report prior-to-posterior change. 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. Efficient inference

A small example makes the mechanism visible. The algorithm updates hidden labels and parameters repeatedly.

A concrete situation

Type and domain assignments stabilize over iterations.

What the mathematics does not license

Convergence does not prove global optimality. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Track traces and restarts. 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. Multi-section sharing

This is where modelling choices begin to affect evidence. Several sections contribute to common type and domain parameters.

A concrete situation

Replicate cortex sections borrow marker evidence.

What the mathematics does not license

Non-comparable sections bias shared estimates. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use leave-one-section-out 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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14. Cell-type proportions

The central mathematical idea is concrete. Posterior compositions summarize each inferred domain.

A concrete situation

A domain contains an estimated immune fraction.

What the mathematics does not license

Compositional uncertainty matters. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Publish credible 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.

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15. Scale-specific outputs

Here the geometry becomes operational. BASS returns cell types, domain labels and their composition link.

A concrete situation

Cellular and regional maps answer different questions.

What the mathematics does not license

One output cannot validate the other. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use separate benchmarks. 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

This step deserves a careful pause. Version features, graphs, priors, type and domain counts, seeds and sections.

A concrete situation

A student calculates one Gaussian-categorical-Potts score.

What the mathematics does not license

A hierarchy is an explanatory model. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Write a BASS evidence ledger. 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

BASS connects two biological scales in one Bayesian hierarchy. At the cell scale, expression features support latent cell-type labels. At the tissue scale, cell types occur in different proportions inside latent spatial domains. A Potts model then makes neighbouring domain labels more likely to agree. The hierarchy explains why a region can contain several cell types while still forming one coherent tissue compartment.

In the documented model, a cell’s expression feature vector follows a multivariate normal distribution with a cell-type-specific mean and shared covariance. Conditional on the spatial domain, the cell type follows a categorical distribution whose probability vector is the domain’s cell-type composition. Domain labels across a section follow a homogeneous Potts model on a neighbourhood graph.

Multiple sections can share cell-type parameters and domain-composition patterns while retaining separate spatial graphs. Borrowing evidence can stabilize rare types and repeated structures, but it is useful only when sections are comparable. If disease, stage and batch are confounded, a shared hierarchy can average away true differences. Posterior uncertainty, prior sensitivity and section holdouts matter.

A small example uses six cells, two expression features, three cell types and two domains. Compute one Gaussian likelihood, multiply it by a domain-specific categorical probability and compare two Potts labelings by counting agreeing edges. Repeating the calculation after changing one cell-type proportion reveals how evidence moves between the cellular and regional scales.

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Frequently asked questions

What is BASS?

BASS is Bayesian Analytics for Spatial Segmentation, a hierarchical method that jointly infers cell types and tissue-scale spatial domains from one or more sections.

What are the two scales?

Cell-type labels describe molecular identity at the cellular scale; spatial-domain labels describe regional organization at the tissue scale.

How are the scales connected?

Every spatial domain has a categorical cell-type composition, so the domain influences the probability that a cell belongs to each type.

What does the Potts model do?

It gives higher prior probability to domain labelings in which neighbouring cells share labels, with strength controlled by an interaction parameter.

Why analyze multiple samples jointly?

Shared parameters can stabilize recurrent cell types and domains, while each section retains its own spatial neighbourhood graph.

Does a posterior domain equal anatomical truth?

No. It is conditional on features, priors, graph, type and domain counts, section comparability and model fit; independent anatomy must validate it.

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Primary source and fact-check note

Zheng Li and Xiang Zhou, “BASS: multi-scale and multi-sample analysis enables accurate cell type clustering and spatial domain detection in spatial transcriptomic studies,” Genome Biology 23, 168 (2022), DOI 10.1186/s13059-022-02734-7. 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

• MENDER multiscale contexts
• BANKSY neighbourhood features
• BayesSpace Potts priors
• 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

Feature audit

Repeat normalized features and rare-marker panels.

Type audit

Inspect posterior probabilities and rare-type recovery.

Composition audit

Report domain cell-type vectors with uncertainty.

Graph audit

Map degrees and physical edge lengths.

Potts audit

Sweep beta at narrow and disconnected structures.

Prior audit

Compare weak and informative priors.

Section audit

Use leave-one-section-out and heterogeneity checks.

Final claim

Cell types and domains are coupled posterior estimates.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Expression feature vectors

Begin with this mechanism: Each cell has normalized molecular features selected for modeling. Work through the concrete case: A cell is represented by ten spatially variable components. Make the limitation visible: Feature selection can omit rare programs. Complete the evidence step: Repeat robust gene sets. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 2: Cell-type labels

Begin with this mechanism: A hidden categorical variable assigns each cell to one type. Work through the concrete case: A neuron-like cell is provisionally type two. Make the limitation visible: A discrete type may hide continua. Complete the evidence step: Inspect posterior probabilities. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 3: Multivariate normal layer

Begin with this mechanism: Expression depends on a cell-type-specific mean and shared covariance. Work through the concrete case: Type two has a distinct two-feature centre. Make the limitation visible: Gaussian tails and shared covariance may misfit. Complete the evidence step: Plot residuals by type. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 4: Shared covariance

Begin with this mechanism: One matrix describes within-type variation across all types. Work through the concrete case: Correlated genes form tilted probability ellipses. Make the limitation visible: Rare types may need different covariance. Complete the evidence step: Compare flexible alternatives. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 5: Spatial-domain labels

Begin with this mechanism: A second hidden variable assigns each cell to a regional domain. Work through the concrete case: Layer four is domain three. Make the limitation visible: Domain count shapes segmentation. Complete the evidence step: Report several counts. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 6: Domain composition vector

Begin with this mechanism: Each domain contains categorical probabilities over cell types. Work through the concrete case: Domain three is 70 per cent neurons and 30 per cent glia. Make the limitation visible: Estimated composition depends on type labels. Complete the evidence step: Propagate uncertainty. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 7: Categorical bridge

Begin with this mechanism: Cell type is drawn conditional on the cell’s domain. Work through the concrete case: A domain rich in type two increases its prior probability. Make the limitation visible: Unusual cells can be overruled by context. Complete the evidence step: Inspect expression-versus-context conflict. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 8: Neighbourhood graph

Begin with this mechanism: Coordinates define edges within every tissue section. Work through the concrete case: A kNN graph links local cells. Make the limitation visible: Density changes physical scale. Complete the evidence step: Map edge lengths. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 9: Potts interaction

Begin with this mechanism: Agreeing neighbouring domain labels increase prior support. Work through the concrete case: A smooth layer beats a checkerboard partition. Make the limitation visible: Strong beta erases thin structures. Complete the evidence step: Publish beta sensitivity. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 10: Joint posterior

Begin with this mechanism: Expression, type, domain composition and graph smoothness combine. Work through the concrete case: One labeling balances molecular fit and spatial coherence. Make the limitation visible: Posterior modes can be multimodal. Complete the evidence step: Use multiple chains or starts. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 11: Bayesian priors

Begin with this mechanism: Parameters receive distributions that regularize sparse evidence. Work through the concrete case: Rare-type proportions shrink toward a prior. Make the limitation visible: Priors can dominate small domains. Complete the evidence step: Report prior-to-posterior change. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 12: Efficient inference

Begin with this mechanism: The algorithm updates hidden labels and parameters repeatedly. Work through the concrete case: Type and domain assignments stabilize over iterations. Make the limitation visible: Convergence does not prove global optimality. Complete the evidence step: Track traces and restarts. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 13: Multi-section sharing

Begin with this mechanism: Several sections contribute to common type and domain parameters. Work through the concrete case: Replicate cortex sections borrow marker evidence. Make the limitation visible: Non-comparable sections bias shared estimates. Complete the evidence step: Use leave-one-section-out checks. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 14: Cell-type proportions

Begin with this mechanism: Posterior compositions summarize each inferred domain. Work through the concrete case: A domain contains an estimated immune fraction. Make the limitation visible: Compositional uncertainty matters. Complete the evidence step: Publish credible intervals. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 15: Scale-specific outputs

Begin with this mechanism: BASS returns cell types, domain labels and their composition link. Work through the concrete case: Cellular and regional maps answer different questions. Make the limitation visible: One output cannot validate the other. Complete the evidence step: Use separate benchmarks. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 16: Governance and learning

Begin with this mechanism: Version features, graphs, priors, type and domain counts, seeds and sections. Work through the concrete case: A student calculates one Gaussian-categorical-Potts score. Make the limitation visible: A hierarchy is an explanatory model. Complete the evidence step: Write a BASS evidence ledger. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.