Chapter map

1. 1. Why markers need a model
2. 2. Raw count inputs
3. 3. Binary marker matrix
4. 4. Latent cell-type variable
5. 5. Library-size factor
6. 6. Gene baseline
7. 7. Positive marker effect
8. 8. Covariate design matrix
9. 9. Expected-count equation
10. 10. Overdispersed count likelihood
11. 11. Class proportions and priors
12. 12. Expectation step
13. 13. Maximisation step
14. 14. Unassigned and omitted types
15. 15. A student mixture project
16. 16. Governance and careers

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1. Why markers need a model

This is where modelling choices begin to affect evidence. Marker lists alone do not quantify expression noise, library size, shared markers or uncertainty. A likelihood connects prior knowledge to counts.

A concrete situation

Two related T-cell types share several markers.

What the mathematics does not license

A rule-based winner hides ambiguous evidence. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Define the generative assumptions first. 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. Raw count inputs

The central mathematical idea is concrete. The model is formulated for nonnegative read or UMI counts rather than arbitrary scaled embeddings.

A concrete situation

A cell has integer counts across marker genes.

What the mathematics does not license

Imputed or negative values violate the contract. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Audit units and integer support. 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. Binary marker matrix

Here the geometry becomes operational. A matrix rho marks whether gene g is a prior marker for type c.

A concrete situation

PTPRC is marked for immune categories.

What the mathematics does not license

Incorrect or incomplete entries bias assignments. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Cite each marker and version the matrix. 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. Latent cell-type variable

This step deserves a careful pause. Each cell has an unobserved categorical variable z indicating its candidate type.

A concrete situation

z equals one of C known types or unassigned.

What the mathematics does not license

The candidate list defines what can be named. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Audit missing types before inference. 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. Library-size factor

The useful question is not only how to calculate, but what the calculation assumes. A per-cell factor scales expected counts so deeper sequencing is not mistaken for marker overexpression.

A concrete situation

A cell with twice the depth has larger expected counts broadly.

What the mathematics does not license

Size estimates can be distorted by composition. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare alternative size 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.

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6. Gene baseline

A small example makes the mechanism visible. Each gene receives a baseline expression term shared across types, separating ordinary abundance from marker lift.

A concrete situation

A common gene has a large baseline but no type effect.

What the mathematics does not license

Baseline assumptions can miss state dependence. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Inspect residuals by type and donor. 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. Positive marker effect

This is where modelling choices begin to affect evidence. When rho equals one, a nonnegative delta raises expected expression for that type; the documented model can constrain a minimum fold increase.

A concrete situation

A marker contributes exp(delta) multiplicatively.

What the mathematics does not license

One fixed relationship may not fit every tissue. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Plot inferred effects 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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8. Covariate design matrix

The central mathematical idea is concrete. Patient, batch or site indicators enter as gene-specific nuisance effects.

A concrete situation

A donor-wide shift is modelled apart from cell type.

What the mathematics does not license

Perfect confounding makes effects unidentifiable. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Cross-tab type and covariates. 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. Expected-count equation

Here the geometry becomes operational. On the log scale, size, baseline, marker lift and covariates add; exponentiation returns a positive mean.

A concrete situation

A toy cell combines four terms.

What the mathematics does not license

A correct mean is not a full distribution. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Calculate one gene–cell expectation. 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. Overdispersed count likelihood

This step deserves a careful pause. Count variance exceeds the Poisson mean in single-cell data, so the likelihood must represent biological and technical dispersion.

A concrete situation

Two cells with the same mean show broad counts.

What the mathematics does not license

Misfit tails distort posterior probabilities. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Check posterior predictive variance and zeros. 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. Class proportions and priors

The useful question is not only how to calculate, but what the calculation assumes. Mixture weights describe expected type frequencies and receive a prior that stabilises estimation.

A concrete situation

A rare type begins with small but nonzero support.

What the mathematics does not license

Strong priors can suppress real rarity. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Run prior-sensitivity analysis. 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. Expectation step

A small example makes the mechanism visible. Given current parameters, the algorithm calculates posterior responsibility of each type for every cell.

A concrete situation

A cell is 0.70 type A and 0.25 type B.

What the mathematics does not license

Responsibilities inherit model misspecification. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Preserve the full vector. 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. Maximisation step

This is where modelling choices begin to affect evidence. Parameters are updated to increase the expected complete-data objective under current responsibilities and constraints.

A concrete situation

Marker effects and covariates move together.

What the mathematics does not license

Local optima and initialisation can matter. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Repeat seeds and inspect objective traces. 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. Unassigned and omitted types

The central mathematical idea is concrete. Cells poorly explained by declared markers can remain unassigned, but an omitted type may still be absorbed by a related known type.

A concrete situation

A missing stromal type shows diffuse probabilities.

What the mathematics does not license

Rejection depends on model and threshold. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Withhold known types intentionally. 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. A student mixture project

Here the geometry becomes operational. Simulate counts from two types, shared markers, library sizes and one batch effect; then alternate posterior and parameter updates.

A concrete situation

Known truth reveals confounding and uncertainty.

What the mathematics does not license

Real data add many genes and types. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Perturb the marker matrix. 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 careers

This step deserves a careful pause. Version counts, markers, priors, size factors, covariates, initialisation, objective and posteriors. This joins probability and biological curation.

A concrete situation

Students can begin with mixtures and log links.

What the mathematics does not license

A posterior assignment proves no function or diagnosis. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Write a probabilistic annotation 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

A CellAssign audit starts with raw count data, a binary marker-gene-by-cell-type matrix, per-cell size factors and optional covariates for patient, batch or site. A probabilistic count model assumes declared markers are overexpressed in their associated types, estimates marker effects and nuisance coefficients, and returns posterior assignment probabilities through expectation–maximisation.

Record count units, marker provenance, marker matrix, size factors, covariate design, known types, unassigned policy, overexpression constraints, priors, initial values, optimisation trace, posterior probabilities and maximum-a-posteriori labels. Markers need not be exclusive, but weak, missing or context-dependent markers can make several assignments plausible.

Validate with simulated truth, sorted populations, platform-held-out data, marker perturbation, batch-confounded designs, posterior calibration and omitted-type challenges. A high assignment probability is meaningful only under a count model and marker specification that fit the data.

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Frequently asked questions

What is CellAssign?

CellAssign is a probabilistic model that uses raw counts and a marker-gene matrix to assign cells to declared types while accounting for size factors and optional covariates.

What is the marker matrix?

Rows are genes, columns are cell types and binary entries declare which genes are expected to be overexpressed for which types.

Must a marker be exclusive?

No. The model allows a marker to be expressed elsewhere; it assumes a positive overexpression effect in the types for which it is declared.

What does the count model separate?

It combines library size, gene baseline, type-specific marker effect and covariates such as patient or batch in the expected count.

What does expectation–maximisation do?

It alternates between estimating assignment probabilities and updating model parameters until a declared convergence rule is reached.

How should CellAssign be validated?

Use sorted or simulated truth, marker perturbation, platform-held-out data, batch-confounded challenges, probability calibration and omitted-type tests.

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Primary source and fact-check note

Allen W. Zhang, Ciara O’Flanagan, Elizabeth A. Chavez et al., “Probabilistic cell-type assignment of single-cell RNA-seq for tumor microenvironment profiling,” Nature Methods 16, 1007–1015 (2019), DOI 10.1038/s41592-019-0529-1. 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

• Garnett marker-seeded classifiers
• CellTypist linear annotation
• CellO ontology classification

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

Marker-provenance audit

Cite tissue, species and evidence for every matrix entry. Record disputed and shared markers rather than hiding them.

Design-rank audit

Cross-tab type, patient, site and batch. Perfect confounding prevents the model from separating biological and nuisance effects.

Posterior-predictive audit

Simulate counts from fitted parameters and compare zeros, means, variances and tails by gene, type and batch.

Marker-perturbation test

Flip, remove and add matrix entries in validation analyses. Stable assignments should not depend on one fragile marker.

Initialisation stability

Repeat EM starts and compare objectives, marker effects, class proportions and posterior assignments. Local optima must remain visible.

Calibration audit

Group posterior probabilities into bins and compare with observed correctness in sorted or held-out data. Domain-specific calibration is essential.

Omitted-type challenge

Remove a true type from the candidate matrix. Measure unassigned outcomes, diffuse posteriors and absorption by related classes.

Final claim

Name counts, marker matrix, priors, covariates, convergence and validation. A posterior is conditional on the model and prior knowledge.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Why markers need a model

Start with the mechanism: Marker lists alone do not quantify expression noise, library size, shared markers or uncertainty. A likelihood connects prior knowledge to counts. The worked case is concrete: Two related T-cell types share several markers. Expose the boundary: A rule-based winner hides ambiguous evidence. Complete this evidence step: Define the generative assumptions first. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 2: Raw count inputs

Start with the mechanism: The model is formulated for nonnegative read or UMI counts rather than arbitrary scaled embeddings. The worked case is concrete: A cell has integer counts across marker genes. Expose the boundary: Imputed or negative values violate the contract. Complete this evidence step: Audit units and integer support. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 3: Binary marker matrix

Start with the mechanism: A matrix rho marks whether gene g is a prior marker for type c. The worked case is concrete: PTPRC is marked for immune categories. Expose the boundary: Incorrect or incomplete entries bias assignments. Complete this evidence step: Cite each marker and version the matrix. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 4: Latent cell-type variable

Start with the mechanism: Each cell has an unobserved categorical variable z indicating its candidate type. The worked case is concrete: z equals one of C known types or unassigned. Expose the boundary: The candidate list defines what can be named. Complete this evidence step: Audit missing types before inference. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 5: Library-size factor

Start with the mechanism: A per-cell factor scales expected counts so deeper sequencing is not mistaken for marker overexpression. The worked case is concrete: A cell with twice the depth has larger expected counts broadly. Expose the boundary: Size estimates can be distorted by composition. Complete this evidence step: Compare alternative size factors. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 6: Gene baseline

Start with the mechanism: Each gene receives a baseline expression term shared across types, separating ordinary abundance from marker lift. The worked case is concrete: A common gene has a large baseline but no type effect. Expose the boundary: Baseline assumptions can miss state dependence. Complete this evidence step: Inspect residuals by type and donor. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 7: Positive marker effect

Start with the mechanism: When rho equals one, a nonnegative delta raises expected expression for that type; the documented model can constrain a minimum fold increase. The worked case is concrete: A marker contributes exp(delta) multiplicatively. Expose the boundary: One fixed relationship may not fit every tissue. Complete this evidence step: Plot inferred effects and uncertainty. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 8: Covariate design matrix

Start with the mechanism: Patient, batch or site indicators enter as gene-specific nuisance effects. The worked case is concrete: A donor-wide shift is modelled apart from cell type. Expose the boundary: Perfect confounding makes effects unidentifiable. Complete this evidence step: Cross-tab type and covariates. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 9: Expected-count equation

Start with the mechanism: On the log scale, size, baseline, marker lift and covariates add; exponentiation returns a positive mean. The worked case is concrete: A toy cell combines four terms. Expose the boundary: A correct mean is not a full distribution. Complete this evidence step: Calculate one gene–cell expectation. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 10: Overdispersed count likelihood

Start with the mechanism: Count variance exceeds the Poisson mean in single-cell data, so the likelihood must represent biological and technical dispersion. The worked case is concrete: Two cells with the same mean show broad counts. Expose the boundary: Misfit tails distort posterior probabilities. Complete this evidence step: Check posterior predictive variance and zeros. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 11: Class proportions and priors

Start with the mechanism: Mixture weights describe expected type frequencies and receive a prior that stabilises estimation. The worked case is concrete: A rare type begins with small but nonzero support. Expose the boundary: Strong priors can suppress real rarity. Complete this evidence step: Run prior-sensitivity analysis. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 12: Expectation step

Start with the mechanism: Given current parameters, the algorithm calculates posterior responsibility of each type for every cell. The worked case is concrete: A cell is 0.70 type A and 0.25 type B. Expose the boundary: Responsibilities inherit model misspecification. Complete this evidence step: Preserve the full vector. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 13: Maximisation step

Start with the mechanism: Parameters are updated to increase the expected complete-data objective under current responsibilities and constraints. The worked case is concrete: Marker effects and covariates move together. Expose the boundary: Local optima and initialisation can matter. Complete this evidence step: Repeat seeds and inspect objective traces. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 14: Unassigned and omitted types

Start with the mechanism: Cells poorly explained by declared markers can remain unassigned, but an omitted type may still be absorbed by a related known type. The worked case is concrete: A missing stromal type shows diffuse probabilities. Expose the boundary: Rejection depends on model and threshold. Complete this evidence step: Withhold known types intentionally. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 15: A student mixture project

Start with the mechanism: Simulate counts from two types, shared markers, library sizes and one batch effect; then alternate posterior and parameter updates. The worked case is concrete: Known truth reveals confounding and uncertainty. Expose the boundary: Real data add many genes and types. Complete this evidence step: Perturb the marker matrix. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.

Workshop 16: Governance and careers

Start with the mechanism: Version counts, markers, priors, size factors, covariates, initialisation, objective and posteriors. This joins probability and biological curation. The worked case is concrete: Students can begin with mixtures and log links. Expose the boundary: A posterior assignment proves no function or diagnosis. Complete this evidence step: Write a probabilistic annotation card. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare features, bins, trees, activations, probabilities, hierarchy edits or labels. Use untouched donors, studies or simulated truth because random cells can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision on separate notebook lines so the conclusion remains auditable.