Chapter map

1. 1. Annotated single-cell input
2. 2. Cell library scaling
3. 3. Type-by-gene rate
4. 4. Gene noise parameter
5. 5. Single-cell likelihood
6. 6. Spatial mixture weight
7. 7. Mixture mean
8. 8. Spatial negative binomial
9. 9. Log-likelihood optimization
10. 10. Maximum a posteriori rule
11. 11. Proportion normalization
12. 12. Two-stage reuse
13. 13. Worked three-gene mixture
14. 14. Known-mixture benchmark
15. 15. Reference-mismatch test
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. Annotated single-cell input

The central mathematical idea is concrete. Raw counts and cell-type labels define the reference population for rate estimation.

A concrete situation

Five immune types are represented across three donors.

What the mathematics does not license

Label errors propagate directly. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Audit labels and markers. 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 library scaling

Here the geometry becomes operational. A cell-specific total-count factor scales the expected counts while preserving type-specific relative rates.

A concrete situation

A cell with twice the library depth has twice the expected counts.

What the mathematics does not license

Depth can correlate with type. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect size factors 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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3. Type-by-gene rate

This step deserves a careful pause. A positive rate parameter captures average expression for gene g in cell type t.

A concrete situation

Cardiomyocytes have a high rate for a contractile gene.

What the mathematics does not license

Average rates hide state variation. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Stratify known states. 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. Gene noise parameter

The useful question is not only how to calculate, but what the calculation assumes. A gene-specific negative binomial parameter controls variability around the scaled mean.

A concrete situation

A bursty gene has broader tails than a stable gene.

What the mathematics does not license

Shared noise across types is an assumption. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Check type-stratified residuals. 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. Single-cell likelihood

A small example makes the mechanism visible. The product of negative binomial probabilities estimates reference rates and noise from raw counts.

A concrete situation

Thousands of cells contribute log-likelihood terms.

What the mathematics does not license

Large groups dominate the objective. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Balance and report group sizes. 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. Spatial mixture weight

This is where modelling choices begin to affect evidence. Every location has one nonnegative weight for each reference cell type.

A concrete situation

A spot has weights 4, 2 and 0.5 for three types.

What the mathematics does not license

Weights are not yet proportions. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Keep the original scale. 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. Mixture mean

The central mathematical idea is concrete. The expected spatial count for a gene is a weighted sum of type-specific rates with a location scale.

A concrete situation

Weights combine three reference rates into one mean.

What the mathematics does not license

Collinear rates create ambiguity. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Measure signature similarity. 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. Spatial negative binomial

Here the geometry becomes operational. Observed spatial counts use the mixture mean and gene noise parameter in a count distribution.

A concrete situation

A mixed mean of 10 produces an overdispersed count.

What the mathematics does not license

Platform noise may differ from scRNA noise. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check technology residuals. 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. Log-likelihood optimization

This step deserves a careful pause. Products become sums of log probabilities, making gradient optimization numerically stable.

A concrete situation

One poor gene contributes a large negative term.

What the mathematics does not license

Local optima and scaling matter. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Repeat seeds and inspect convergence. 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. Maximum a posteriori rule

The useful question is not only how to calculate, but what the calculation assumes. Optional priors regularize positive mixture weights, and optimization returns the mode rather than the full posterior.

A concrete situation

A weak prior discourages extreme allocations.

What the mathematics does not license

A mode omits uncertainty shape. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use bootstrap or sensitivity 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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11. Proportion normalization

A small example makes the mechanism visible. Dividing each weight by the location total creates fractions that sum to one.

A concrete situation

Weights 4, 2 and 0.5 become 0.615, 0.308 and 0.077.

What the mathematics does not license

RNA proportions differ from cell fractions. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Label the quantity precisely. 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. Two-stage reuse

This is where modelling choices begin to affect evidence. Once reference parameters are fitted, the same signatures can deconvolve additional spatial sections.

A concrete situation

One atlas is applied to two adjacent tissue sections.

What the mathematics does not license

Reference mismatch can increase across sections. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Hold out complete sections. 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. Worked three-gene mixture

The central mathematical idea is concrete. Use two type-specific rate vectors, two weights and a depth factor to calculate a spatial mean vector.

A concrete situation

A 70:30 RNA mixture predicts three gene means.

What the mathematics does not license

Rates and depth are simplified. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Add a third type and noise. 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. Known-mixture benchmark

Here the geometry becomes operational. Synthetic or experimentally mixed samples with known proportions test recovery and calibration.

A concrete situation

True 60:40 mixtures are repeatedly simulated.

What the mathematics does not license

Synthetic data can favour the model. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Include real orthogonal measurements. 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. Reference-mismatch test

This step deserves a careful pause. Remove one type or donor from the reference and observe forced allocation and residuals.

A concrete situation

A missing endothelial subtype is absorbed into another vascular type.

What the mathematics does not license

Good likelihood can hide wrong identity. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Publish absent-type challenges. 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 raw counts, labels, size factors, rates, noise, weights, optimizer and proportions. This joins mixtures and inference.

A concrete situation

Students can begin with weighted averages and log likelihood.

What the mathematics does not license

A proportion is conditional on the reference. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Write a deconvolution 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

A Stereoscope audit uses two linked negative binomial models. First, annotated single-cell counts estimate gene-specific expression rates for each cell type and gene-level noise parameters. Second, every spatial location is treated as a weighted mixture of those rates. Maximum-likelihood or maximum-a-posteriori optimization finds nonnegative mixture weights, which are normalized when proportions are needed.

Record raw count matrices, cell-type labels, shared genes, library-size factors, single-cell rate parameters, gene-specific success or dispersion parameters, spatial weights, priors, optimizer, epochs, convergence and proportion normalization. The two-stage design allows the learned reference parameters to be reused, but it also makes reference mismatch and label quality central limitations.

Validate with known mixtures, complete-location holdouts, marker-free and selected-gene comparisons, alternative references, rare-type challenges, residual count maps and external histology or imaging. A normalized weight is an inferred RNA-composition proportion under the model; it need not equal a physical fraction of cells when cell sizes and RNA content differ.

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Frequently asked questions

What is Stereoscope?

Stereoscope is a probabilistic spatial deconvolution method that learns cell-type-specific expression parameters from annotated single-cell counts and estimates mixture weights for spatial locations.

What are its two stages?

The first stage estimates negative binomial parameters for each gene and cell type. The second combines those parameters with nonnegative weights to explain each spatial count profile.

Why use raw counts?

The likelihood is defined for discrete counts and includes library-size scaling. Transforming counts first can break the stated observation model.

What is MAP estimation?

Maximum a posteriori estimation finds parameter values that maximize likelihood times prior density. Without a prior, the corresponding procedure is maximum likelihood.

How are proportions obtained?

The fitted nonnegative cell-type weights at a location are normalized to sum to one. They describe inferred RNA mixture composition under the model.

Are proportions physical cell fractions?

Not necessarily. Cell types differ in RNA content, size and capture efficiency, and the reference may not match the tissue. Imaging or known mixtures are needed for physical validation.

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Primary source and fact-check note

Alma Andersson, Joseph F. Bergenstråhle, Michaela Asp et al., “Single-cell and spatial transcriptomics enables probabilistic inference of cell type topography,” Communications Biology 3, 565 (2020), DOI 10.1038/s42003-020-01247-y. 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

• PeakVI probability modelling
• DestVI continuous cell states
• scVI negative binomial counts
• 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

Count audit

Confirm raw integer counts and documented library scaling in both modalities.

Label audit

Test cell-type labels, imbalance and donor coverage before rate estimation.

Rate audit

Inspect type-by-gene rates and gene noise with residual plots.

Weight audit

Preserve raw weights before proportion normalization and report both meanings.

Mixture challenge

Use known mixtures and collinear signatures to test separability.

Reference swap

Repeat with plausible alternative atlases and donors.

External check

Compare proportions with histology, imaging or targeted assays.

Final claim

Name rates, noise and optimizer. Normalized RNA weights are not guaranteed cell fractions.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Annotated single-cell input

Begin with this mechanism: Raw counts and cell-type labels define the reference population for rate estimation. Work through the concrete case: Five immune types are represented across three donors. Make the limitation visible: Label errors propagate directly. Complete the evidence step: Audit labels and markers. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 2: Cell library scaling

Begin with this mechanism: A cell-specific total-count factor scales the expected counts while preserving type-specific relative rates. Work through the concrete case: A cell with twice the library depth has twice the expected counts. Make the limitation visible: Depth can correlate with type. Complete the evidence step: Inspect size factors by type. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 3: Type-by-gene rate

Begin with this mechanism: A positive rate parameter captures average expression for gene g in cell type t. Work through the concrete case: Cardiomyocytes have a high rate for a contractile gene. Make the limitation visible: Average rates hide state variation. Complete the evidence step: Stratify known states. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 4: Gene noise parameter

Begin with this mechanism: A gene-specific negative binomial parameter controls variability around the scaled mean. Work through the concrete case: A bursty gene has broader tails than a stable gene. Make the limitation visible: Shared noise across types is an assumption. Complete the evidence step: Check type-stratified residuals. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 5: Single-cell likelihood

Begin with this mechanism: The product of negative binomial probabilities estimates reference rates and noise from raw counts. Work through the concrete case: Thousands of cells contribute log-likelihood terms. Make the limitation visible: Large groups dominate the objective. Complete the evidence step: Balance and report group sizes. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 6: Spatial mixture weight

Begin with this mechanism: Every location has one nonnegative weight for each reference cell type. Work through the concrete case: A spot has weights 4, 2 and 0.5 for three types. Make the limitation visible: Weights are not yet proportions. Complete the evidence step: Keep the original scale. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 7: Mixture mean

Begin with this mechanism: The expected spatial count for a gene is a weighted sum of type-specific rates with a location scale. Work through the concrete case: Weights combine three reference rates into one mean. Make the limitation visible: Collinear rates create ambiguity. Complete the evidence step: Measure signature similarity. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 8: Spatial negative binomial

Begin with this mechanism: Observed spatial counts use the mixture mean and gene noise parameter in a count distribution. Work through the concrete case: A mixed mean of 10 produces an overdispersed count. Make the limitation visible: Platform noise may differ from scRNA noise. Complete the evidence step: Check technology residuals. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 9: Log-likelihood optimization

Begin with this mechanism: Products become sums of log probabilities, making gradient optimization numerically stable. Work through the concrete case: One poor gene contributes a large negative term. Make the limitation visible: Local optima and scaling matter. Complete the evidence step: Repeat seeds and inspect convergence. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 10: Maximum a posteriori rule

Begin with this mechanism: Optional priors regularize positive mixture weights, and optimization returns the mode rather than the full posterior. Work through the concrete case: A weak prior discourages extreme allocations. Make the limitation visible: A mode omits uncertainty shape. Complete the evidence step: Use bootstrap or sensitivity checks. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 11: Proportion normalization

Begin with this mechanism: Dividing each weight by the location total creates fractions that sum to one. Work through the concrete case: Weights 4, 2 and 0.5 become 0.615, 0.308 and 0.077. Make the limitation visible: RNA proportions differ from cell fractions. Complete the evidence step: Label the quantity precisely. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 12: Two-stage reuse

Begin with this mechanism: Once reference parameters are fitted, the same signatures can deconvolve additional spatial sections. Work through the concrete case: One atlas is applied to two adjacent tissue sections. Make the limitation visible: Reference mismatch can increase across sections. Complete the evidence step: Hold out complete sections. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 13: Worked three-gene mixture

Begin with this mechanism: Use two type-specific rate vectors, two weights and a depth factor to calculate a spatial mean vector. Work through the concrete case: A 70:30 RNA mixture predicts three gene means. Make the limitation visible: Rates and depth are simplified. Complete the evidence step: Add a third type and noise. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 14: Known-mixture benchmark

Begin with this mechanism: Synthetic or experimentally mixed samples with known proportions test recovery and calibration. Work through the concrete case: True 60:40 mixtures are repeatedly simulated. Make the limitation visible: Synthetic data can favour the model. Complete the evidence step: Include real orthogonal measurements. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 15: Reference-mismatch test

Begin with this mechanism: Remove one type or donor from the reference and observe forced allocation and residuals. Work through the concrete case: A missing endothelial subtype is absorbed into another vascular type. Make the limitation visible: Good likelihood can hide wrong identity. Complete the evidence step: Publish absent-type challenges. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.

Workshop 16: Governance and learning

Begin with this mechanism: Version raw counts, labels, size factors, rates, noise, weights, optimizer and proportions. This joins mixtures and inference. Work through the concrete case: Students can begin with weighted averages and log likelihood. Make the limitation visible: A proportion is conditional on the reference. Complete the evidence step: Write a deconvolution ledger. Write the matrices, count distribution, mixture, prior, loss function or mapping rule with named dimensions and units. Calculate one small example by hand, then change one assumption and explain which output moves and why. Use complete donors, tissue sections, spatial locations or gene panels outside tuning when testing transfer, because random rows from one specimen are not independent biological evidence. Keep observation, fitted parameter, prediction, interpretation and decision on separate notebook lines so a student, parent or reviewer can see exactly what the mathematics establishes.