WHY MATHEMATICS?

Why Mathematics? | STdeconvolve Reference-Free Cell Types: Overdispersed Genes, Latent Dirichlet Allocation, Perplexity and Topic Proportions

How STdeconvolve turns spatial count pixels into documents, genes into words and latent topics into putative cell types without an external single-cell reference.

Why mathematics is important becomes easier to see when a real decision depends on it. This guide follows the mechanism from definitions and dimensions to a worked calculation, validation, practical learning and responsible use. It keeps factual evidence separate from interpretation and treats every illustrative scenario as an example rather than a promise of results.

Choose a reading route

Select informative genes using detection and overdispersion
Represent every pixel as a mixture of latent topics
Use perplexity and rare-topic counts to guide K
Interpret topics only after stability and external validation

Chapter map

1. Spatial count matrix
2. Detection-frequency filter
3. Expected mean-variance trend
4. Overdispersion statistic
5. Top-feature selection
6. Topic number K
7. Dirichlet prior alpha
8. Pixel topic vector theta
9. Topic gene vector beta
10. Generative count step
11. Likelihood
12. Perplexity
13. Rare-topic count
14. Knee or inflection
15. Topic annotation
16. Stability and validation

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

1. Spatial count matrix

Here the geometry becomes operational. Rows are pixels and columns are genes.

A concrete situation

D equals 2,000 pixels and N equals 15,000 genes.

What the mathematics does not license

Counts mix several cells and technical effects. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Retain integer 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.

2. Detection-frequency filter

This step deserves a careful pause. Genes outside declared pixel-frequency bounds are removed.

A concrete situation

A gene in 2% of pixels is excluded.

What the mathematics does not license

Rare markers can disappear. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Sweep lower and upper bounds. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

3. Expected mean-variance trend

The useful question is not only how to calculate, but what the calculation assumes. A baseline predicts variance from mean expression.

A concrete situation

Gene g has expected variance six.

What the mathematics does not license

Trend choice affects overdispersion. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Plot model 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.

4. Overdispersion statistic

A small example makes the mechanism visible. Observed variance above expectation identifies features.

A concrete situation

Variance ten minus expected six is positive.

What the mathematics does not license

Regional gradients may mimic cell types. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Inspect spatial and technical 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.

Did You Know? In this chapter, the most important number is useful only together with its construction rule. Changing scaling, sample membership or validation design can change the numerical answer without changing the arithmetic.

5. Top-feature selection

This is where modelling choices begin to affect evidence. A ranked subset enters the topic model.

A concrete situation

The top 1,000 genes are retained.

What the mathematics does not license

A cutoff changes identifiability. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Repeat across feature 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.

6. Topic number K

The central mathematical idea is concrete. K declares how many latent profiles will be fitted.

A concrete situation

Models test K from two to fifteen.

What the mathematics does not license

The biological count is not directly observed. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

7. Dirichlet prior alpha

Here the geometry becomes operational. A prior shapes pixel-topic mixtures.

A concrete situation

Alpha equals 50 divided by K by default.

What the mathematics does not license

The prior can favour diffuse mixtures. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

8. Pixel topic vector theta

This step deserves a careful pause. Nonnegative probabilities sum to one per pixel.

A concrete situation

Theta-d is [0.7,0.2,0.1].

What the mathematics does not license

Topic fraction is not necessarily cell fraction. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

9. Topic gene vector beta

The useful question is not only how to calculate, but what the calculation assumes. Each topic is a gene-probability distribution.

A concrete situation

Beta-k sums to one across genes.

What the mathematics does not license

Highly expressed genes can dominate. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

10. Generative count step

A small example makes the mechanism visible. Each molecule selects a topic then a gene.

A concrete situation

One count is assigned through z to gene n.

What the mathematics does not license

Counts are treated as exchangeable within a pixel. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

11. Likelihood

This is where modelling choices begin to affect evidence. All molecule probabilities combine into model fit.

A concrete situation

Log likelihood is minus 82,000.

What the mathematics does not license

Larger datasets change scale. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

12. Perplexity

The central mathematical idea is concrete. Exponentiated negative log likelihood per count gives a fit score.

A concrete situation

Perplexity falls from 620 to 510.

What the mathematics does not license

It often improves as K grows. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use it with complexity diagnostics. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

13. Rare-topic count

Here the geometry becomes operational. Low mean theta identifies tiny inferred components.

A concrete situation

Two topics average below 5%.

What the mathematics does not license

Real rare cells and spurious splits overlap. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

14. Knee or inflection

This step deserves a careful pause. Curvature highlights diminishing returns.

A concrete situation

The second derivative peaks near K equals eight.

What the mathematics does not license

Knees depend on the tested range. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

15. Topic annotation

The useful question is not only how to calculate, but what the calculation assumes. Beta profiles are matched to markers or orthogonal evidence.

A concrete situation

Topic 3 enriches endothelial genes.

What the mathematics does not license

Post-hoc labels can be circular. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use held-out markers and anatomy. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

16. Stability and validation

A small example makes the mechanism visible. Seeds, features and K are perturbed.

A concrete situation

A topic recurs in 18 of 20 fits.

What the mathematics does not license

Stable technical structure can persist. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Validate with independent 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.

A seven-day practice lab

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

STdeconvolve is reference-free: it uses only the spatial gene-count matrix and treats each multi-cellular pixel like a document. Genes take the role of words, latent cell types take the role of topics and each pixel has a probability mixture over those topics. The model estimates theta, a pixel-by-cell-type proportion matrix, and beta, a cell-type-by-gene probability matrix.

The method first filters genes detected in too few or too many pixels and selects significantly overdispersed genes. This is a mathematical bet that cell-type-specific genes vary more across pixels than expected. The filter can sharpen latent topics, but it can also remove broadly expressed identity markers or retain state, region and technical effects that are not separate cell types.

Latent Dirichlet allocation requires the number of topics K in advance. STdeconvolve fits several K values, computes perplexity from the fitted likelihood per observed count and monitors how many inferred topics have very low mean proportions. Lower perplexity improves data fit, yet it often keeps falling as K grows, while rare or unstable topics multiply. The knee and rare-topic count guide rather than prove the biological K.

Because topic labels are not supplied, beta profiles must be interpreted after fitting using marker enrichment, anatomy or independent measurements. Reference-free does not mean assumption-free. Dirichlet priors, feature selection, K, random initialization and the bag-of-counts assumptions determine what patterns can appear. Repeated fits and external validation are essential before calling a topic a cell type.

Frequently asked questions

What is STdeconvolve?

STdeconvolve is an unsupervised, reference-free method that applies latent Dirichlet allocation to spatial gene-count pixels.

What are theta and beta?

Theta gives each pixel’s topic proportions; beta gives each topic’s probability distribution over genes.

Why select overdispersed genes?

Genes varying more than expected across pixels are more likely to separate latent transcriptional populations.

Why must K be chosen?

LDA needs the number of topics before fitting; different K values produce different decompositions.

What does perplexity say?

Perplexity summarizes how well the fitted probabilistic model predicts the observed counts; lower values indicate better fit under the model.

Does a latent topic automatically equal a cell type?

No. It may reflect a cell type, state, region or technical effect and requires interpretation and validation.

Primary source and fact-check note

Brendan F. Miller, Feiyang Huang, Lyla Atta, Arpan Sahoo and Jean Fan, “Reference-free cell type deconvolution of multi-cellular pixel-resolution spatially resolved transcriptomics data,” Nature Communications 13, 2339 (2022), DOI 10.1038/s41467-022-30033-z. 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

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

Retain integer counts and library sizes.

Feature audit

Publish detection and overdispersion filters.

Prior audit

Record alpha, initialization and convergence.

K audit

Show perplexity and rare-topic curves for every K.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Stability audit

Match topics across seeds and feature sets.

Annotation audit

Use held-out markers and anatomy.

Fraction audit

Avoid equating topic mass with physical cell count.

Final claim

Reference-free models still depend on assumptions and K.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Spatial count matrix

Begin with this mechanism: Rows are pixels and columns are genes. Recreate the concrete case: D equals 2,000 pixels and N equals 15,000 genes. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Counts mix several cells and technical effects. Complete the transfer step: Retain integer counts. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 2: Detection-frequency filter

Begin with this mechanism: Genes outside declared pixel-frequency bounds are removed. Recreate the concrete case: A gene in 2% of pixels is excluded. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Rare markers can disappear. Complete the transfer step: Sweep lower and upper bounds. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 3: Expected mean-variance trend

Begin with this mechanism: A baseline predicts variance from mean expression. Recreate the concrete case: Gene g has expected variance six. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Trend choice affects overdispersion. Complete the transfer step: Plot model residuals. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 4: Overdispersion statistic

Begin with this mechanism: Observed variance above expectation identifies features. Recreate the concrete case: Variance ten minus expected six is positive. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Regional gradients may mimic cell types. Complete the transfer step: Inspect spatial and technical covariates. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 5: Top-feature selection

Begin with this mechanism: A ranked subset enters the topic model. Recreate the concrete case: The top 1,000 genes are retained. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: A cutoff changes identifiability. Complete the transfer step: Repeat across feature counts. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 6: Topic number K

Begin with this mechanism: K declares how many latent profiles will be fitted. Recreate the concrete case: Models test K from two to fifteen. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: The biological count is not directly observed. Complete the transfer step: Compare several diagnostics. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 7: Dirichlet prior alpha

Begin with this mechanism: A prior shapes pixel-topic mixtures. Recreate the concrete case: Alpha equals 50 divided by K by default. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: The prior can favour diffuse mixtures. Complete the transfer step: Test alternative concentration. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 8: Pixel topic vector theta

Begin with this mechanism: Nonnegative probabilities sum to one per pixel. Recreate the concrete case: Theta-d is [0.7,0.2,0.1]. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Topic fraction is not necessarily cell fraction. Complete the transfer step: Use putative language. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 9: Topic gene vector beta

Begin with this mechanism: Each topic is a gene-probability distribution. Recreate the concrete case: Beta-k sums to one across genes. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Highly expressed genes can dominate. Complete the transfer step: Inspect marker enrichment. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 10: Generative count step

Begin with this mechanism: Each molecule selects a topic then a gene. Recreate the concrete case: One count is assigned through z to gene n. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Counts are treated as exchangeable within a pixel. Complete the transfer step: State the bag-of-counts assumption. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 11: Likelihood

Begin with this mechanism: All molecule probabilities combine into model fit. Recreate the concrete case: Log likelihood is minus 82,000. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Larger datasets change scale. Complete the transfer step: Normalize comparisons. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 12: Perplexity

Begin with this mechanism: Exponentiated negative log likelihood per count gives a fit score. Recreate the concrete case: Perplexity falls from 620 to 510. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: It often improves as K grows. Complete the transfer step: Use it with complexity diagnostics. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 13: Rare-topic count

Begin with this mechanism: Low mean theta identifies tiny inferred components. Recreate the concrete case: Two topics average below 5%. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Real rare cells and spurious splits overlap. Complete the transfer step: Inspect biology and stability. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 14: Knee or inflection

Begin with this mechanism: Curvature highlights diminishing returns. Recreate the concrete case: The second derivative peaks near K equals eight. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Knees depend on the tested range. Complete the transfer step: Publish the entire curve. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 15: Topic annotation

Begin with this mechanism: Beta profiles are matched to markers or orthogonal evidence. Recreate the concrete case: Topic 3 enriches endothelial genes. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Post-hoc labels can be circular. Complete the transfer step: Use held-out markers and anatomy. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 16: Stability and validation

Begin with this mechanism: Seeds, features and K are perturbed. Recreate the concrete case: A topic recurs in 18 of 20 fits. Write pixel counts, detection rates, means, variances, overdispersion scores, Dirichlet parameters, theta values, beta values, likelihoods, perplexities and topic stability with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Stable technical structure can persist. Complete the transfer step: Validate with independent measurements. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.