WHY MATHEMATICS?

Why Mathematics? | SciBet for Portable Cell Typing: Entropy Differences, Arithmetic–Geometric Means, Multinomial Likelihoods and KL Divergence

How SciBet turns labelled single-cell expression into a compact probability model: an entropy test chooses informative genes, class profiles become multinomial parameters and each query cell is matched through likelihood or equivalent KL divergence.

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

Measure between-type expression differences with the E-test
Read the arithmetic–geometric mean ratio as an information score
Build smoothed multinomial class profiles
Classify by log-likelihood, KL divergence and honest reference coverage

Chapter map

1. Why a portable probability model helps
2. Class-wise mean expression
3. Differential entropy premise
4. Null group and arithmetic mean
5. Geometric mean
6. E-test statistic
7. AM–GM inequality
8. Selecting top genes
9. Class probability vector
10. Laplace smoothing
11. Multinomial likelihood
12. Log-likelihood arithmetic
13. KL-divergence equivalence
14. Closed-set limitation
15. A student probability project
16. Governance and careers

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

1. Why a portable probability model helps

Here the geometry becomes operational. Large atlases make reference labels available, but query studies differ in depth, platform and cell composition. A compact probability model makes every classification calculation visible.

A concrete situation

One immune reference must label a new donor study.

What the mathematics does not license

Portability fails when query biology is missing from the reference. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Map reference coverage before training. 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. Class-wise mean expression

This step deserves a careful pause. For gene i and class j, SciBet starts from the mean expression Xij across labelled cells in that class.

A concrete situation

Three class means are 2, 2 and 8.

What the mathematics does not license

Means hide dropout and within-class spread. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Report sample size and dispersion beside each mean. 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. Differential entropy premise

The useful question is not only how to calculate, but what the calculation assumes. Under the paper’s Poisson–Gamma assumption, the class-specific differential entropy is written as log Xij plus a gene-specific constant.

A concrete situation

The same constant appears in every class for one gene.

What the mathematics does not license

The assumption is a modelling device, not a universal biological law. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Check count and transformed-data contracts. 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. Null group and arithmetic mean

A small example makes the mechanism visible. The null group averages the class means, giving the class-level arithmetic mean for a gene.

A concrete situation

For 2, 2 and 8, AM is 4.

What the mathematics does not license

Equal weighting of classes differs from cell-weighted averaging. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

State the averaging convention. 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. Geometric mean

This is where modelling choices begin to affect evidence. The geometric mean multiplies the positive class means and takes the nth root.

A concrete situation

For 2, 2 and 8, GM is the cube root of 32.

What the mathematics does not license

Zeros require a declared treatment before logarithms. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Audit pseudocounts and numerical 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.

6. E-test statistic

The central mathematical idea is concrete. The total entropy difference simplifies to n times log(AM/GM). It is zero for identical class means and grows with nonuniformity.

A concrete situation

Using AM 4 and GM about 3.175 gives a positive score.

What the mathematics does not license

A high score can arise from one extreme class or noise. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Inspect class-wise contributions. 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. AM–GM inequality

Here the geometry becomes operational. AM at least GM guarantees a nonnegative statistic when inputs meet the assumptions. This gives the feature score a clear zero baseline.

A concrete situation

Equal means make the ratio exactly one.

What the mathematics does not license

Finite-sample estimates can still be unstable. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Bootstrap the class means. 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. Selecting top genes

This step deserves a careful pause. Genes are ranked by entropy difference and the published default retains 500 informative genes.

A concrete situation

A marker-like gene outranks a uniform housekeeping gene.

What the mathematics does not license

A fixed count is a tuning choice. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Plot held-out performance and stability versus feature count. 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. Class probability vector

The useful question is not only how to calculate, but what the calculation assumes. For each type, the selected gene means are normalised across genes to form probabilities that sum to one.

A concrete situation

A three-gene profile becomes 0.2, 0.3 and 0.5.

What the mathematics does not license

Compositional normalisation couples every probability. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Check the sum and sensitivity to dominant genes. 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. Laplace smoothing

A small example makes the mechanism visible. Adding one to every selected-gene mean before normalising keeps every estimated probability positive.

A concrete situation

An unseen training gene receives small nonzero support.

What the mathematics does not license

Smoothing strength matters most for shallow classes. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare additive constants inside validation. 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. Multinomial likelihood

This is where modelling choices begin to affect evidence. For a query expression vector, the likelihood multiplies class gene probabilities raised to query counts, with a combinatorial factor common across classes.

A concrete situation

Two observed molecules in gene A contribute pA squared.

What the mathematics does not license

Independence and fixed-total assumptions simplify biology. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use log-likelihood and inspect residual structure. 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. Log-likelihood arithmetic

The central mathematical idea is concrete. Products of tiny probabilities become sums of query counts times log probabilities, avoiding underflow and revealing gene contributions.

A concrete situation

A marker count adds strong evidence to one class.

What the mathematics does not license

One highly expressed gene can dominate. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Decompose score by gene. 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. KL-divergence equivalence

Here the geometry becomes operational. After converting query counts to proportions, maximising likelihood is equivalent to minimising KL divergence from the query distribution to a class profile.

A concrete situation

The nearest profile in KL receives the label.

What the mathematics does not license

KL is asymmetric and punishes missing support strongly. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Calculate both directions in a toy example. 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. Closed-set limitation

This step deserves a careful pause. Maximum likelihood returns the best offered class even when every class fits badly. A margin or reference-distance rule must be validated before rejection.

A concrete situation

An absent stromal type is forced toward the least-wrong immune type.

What the mathematics does not license

A large winner score is not enough without a baseline. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Withhold one known 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.

15. A student probability project

The useful question is not only how to calculate, but what the calculation assumes. Compute AM, GM, entropy differences, smoothed class probabilities and log-likelihoods for a tiny labelled table.

A concrete situation

Hand calculations expose every assumption.

What the mathematics does not license

Real atlases add thousands of sparse genes. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Repeat after a platform-scale shift. 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. Governance and careers

A small example makes the mechanism visible. Version labels, class means, E-test scores, selected genes, smoothing, likelihoods and validation. This connects information theory, probability and bioinformatics.

A concrete situation

Students can begin with means, logarithms and proportions.

What the mathematics does not license

A cell label does not establish mechanism or diagnosis. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Write a portable-classifier model 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.

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

A SciBet audit begins with a labelled reference matrix, declared cell-type labels and an untouched query study. It computes class-wise mean expression, ranks genes with an entropy-difference statistic, keeps the most informative genes, estimates a multinomial expression profile for every class with Laplace smoothing and assigns each query cell by maximum likelihood.

Record expression units, scaling and log transformation, class means, the arithmetic and geometric means used by the E-test, the selected-gene count, smoothing rule, class-specific probability vectors, query log-likelihoods and tie policy. The maximum-likelihood winner is always relative to the offered classes, so a closed reference cannot by itself prove that a query belongs to a known type.

Validate on whole donors, platforms and tissues withheld from training; report per-class precision and recall, likelihood margins, calibration or rejection rules, feature stability and absent-class challenges. A portable classifier must remain honest about reference coverage, sparsity and domain shift.

Frequently asked questions

What is SciBet?

SciBet is a supervised single-cell identifier that selects informative genes with an entropy-difference test and models each cell type with a multinomial gene-expression profile.

What does the E-test measure?

For each gene it compares class-level arithmetic and geometric means. The published statistic is n times the logarithm of their ratio, so uniform class means give zero and uneven class means give a positive value.

Why does the arithmetic mean exceed the geometric mean?

For nonnegative values the AM–GM inequality gives AM at least GM. Equality occurs when all class means are equal, which makes the SciBet entropy-difference statistic zero.

Why use Laplace smoothing?

Adding one before normalising class mean expression prevents a zero estimated gene probability from making a whole query likelihood zero.

How is a cell assigned?

SciBet selects the class with the largest multinomial likelihood; after removing terms common to all classes, this is equivalent to the smallest KL divergence between the query gene proportions and a class profile.

Can SciBet identify a cell type absent from the reference?

The closed maximum-likelihood rule always offers a known-class winner. Unknown-type handling therefore needs an explicit rejection or downstream review policy and an absent-class validation test.

Primary source and fact-check note

Chuan Li, Bolun Liu, Biao Kang et al., “SciBet as a portable and fast single cell type identifier,” Nature Communications 11, 1818 (2020), DOI 10.1038/s41467-020-15523-2. 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

E-test arithmetic audit

For every selected gene, store class means, arithmetic mean, geometric mean, log ratio and bootstrap interval. Verify nonnegativity and document zero handling.

Feature-count challenge

Repeat with several selected-gene counts. Report stability, per-class metrics and whether the same biological programmes persist.

Probability-vector audit

Check positivity and a sum of one after Laplace smoothing for every class profile. Record classes most changed by smoothing.

Likelihood decomposition

Save per-gene contributions to each query log-likelihood. Large contributions reveal both informative markers and fragile domination.

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.

Prior-sensitivity test

Compare maximum likelihood with declared class priors where appropriate. Prevalence assumptions should not remain hidden.

Absent-class test

Remove a known class and quantify forced assignments, likelihood margins and any rejection policy. Closed-set success is insufficient.

Platform holdout

Keep complete technologies and studies out of feature selection and validation. Random cells share batch structure.

Final claim

Name reference, features, smoothing, priors, likelihood rule and held-out domains. A portable label is conditional evidence.

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: Why a portable probability model helps

Start with the mechanism: Large atlases make reference labels available, but query studies differ in depth, platform and cell composition. A compact probability model makes every classification calculation visible. The worked case is concrete: One immune reference must label a new donor study. Expose the boundary: Portability fails when query biology is missing from the reference. Complete this evidence step: Map reference coverage before training. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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: Class-wise mean expression

Start with the mechanism: For gene i and class j, SciBet starts from the mean expression Xij across labelled cells in that class. The worked case is concrete: Three class means are 2, 2 and 8. Expose the boundary: Means hide dropout and within-class spread. Complete this evidence step: Report sample size and dispersion beside each mean. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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: Differential entropy premise

Start with the mechanism: Under the paper’s Poisson–Gamma assumption, the class-specific differential entropy is written as log Xij plus a gene-specific constant. The worked case is concrete: The same constant appears in every class for one gene. Expose the boundary: The assumption is a modelling device, not a universal biological law. Complete this evidence step: Check count and transformed-data contracts. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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: Null group and arithmetic mean

Start with the mechanism: The null group averages the class means, giving the class-level arithmetic mean for a gene. The worked case is concrete: For 2, 2 and 8, AM is 4. Expose the boundary: Equal weighting of classes differs from cell-weighted averaging. Complete this evidence step: State the averaging convention. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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.

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: Geometric mean

Start with the mechanism: The geometric mean multiplies the positive class means and takes the nth root. The worked case is concrete: For 2, 2 and 8, GM is the cube root of 32. Expose the boundary: Zeros require a declared treatment before logarithms. Complete this evidence step: Audit pseudocounts and numerical stability. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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: E-test statistic

Start with the mechanism: The total entropy difference simplifies to n times log(AM/GM). It is zero for identical class means and grows with nonuniformity. The worked case is concrete: Using AM 4 and GM about 3.175 gives a positive score. Expose the boundary: A high score can arise from one extreme class or noise. Complete this evidence step: Inspect class-wise contributions. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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: AM–GM inequality

Start with the mechanism: AM at least GM guarantees a nonnegative statistic when inputs meet the assumptions. This gives the feature score a clear zero baseline. The worked case is concrete: Equal means make the ratio exactly one. Expose the boundary: Finite-sample estimates can still be unstable. Complete this evidence step: Bootstrap the class means. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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: Selecting top genes

Start with the mechanism: Genes are ranked by entropy difference and the published default retains 500 informative genes. The worked case is concrete: A marker-like gene outranks a uniform housekeeping gene. Expose the boundary: A fixed count is a tuning choice. Complete this evidence step: Plot held-out performance and stability versus feature count. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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.

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: Class probability vector

Start with the mechanism: For each type, the selected gene means are normalised across genes to form probabilities that sum to one. The worked case is concrete: A three-gene profile becomes 0.2, 0.3 and 0.5. Expose the boundary: Compositional normalisation couples every probability. Complete this evidence step: Check the sum and sensitivity to dominant genes. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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: Laplace smoothing

Start with the mechanism: Adding one to every selected-gene mean before normalising keeps every estimated probability positive. The worked case is concrete: An unseen training gene receives small nonzero support. Expose the boundary: Smoothing strength matters most for shallow classes. Complete this evidence step: Compare additive constants inside validation. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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: Multinomial likelihood

Start with the mechanism: For a query expression vector, the likelihood multiplies class gene probabilities raised to query counts, with a combinatorial factor common across classes. The worked case is concrete: Two observed molecules in gene A contribute pA squared. Expose the boundary: Independence and fixed-total assumptions simplify biology. Complete this evidence step: Use log-likelihood and inspect residual structure. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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: Log-likelihood arithmetic

Start with the mechanism: Products of tiny probabilities become sums of query counts times log probabilities, avoiding underflow and revealing gene contributions. The worked case is concrete: A marker count adds strong evidence to one class. Expose the boundary: One highly expressed gene can dominate. Complete this evidence step: Decompose score by gene. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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.

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: KL-divergence equivalence

Start with the mechanism: After converting query counts to proportions, maximising likelihood is equivalent to minimising KL divergence from the query distribution to a class profile. The worked case is concrete: The nearest profile in KL receives the label. Expose the boundary: KL is asymmetric and punishes missing support strongly. Complete this evidence step: Calculate both directions in a toy example. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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: Closed-set limitation

Start with the mechanism: Maximum likelihood returns the best offered class even when every class fits badly. A margin or reference-distance rule must be validated before rejection. The worked case is concrete: An absent stromal type is forced toward the least-wrong immune type. Expose the boundary: A large winner score is not enough without a baseline. Complete this evidence step: Withhold one known type. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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 probability project

Start with the mechanism: Compute AM, GM, entropy differences, smoothed class probabilities and log-likelihoods for a tiny labelled table. The worked case is concrete: Hand calculations expose every assumption. Expose the boundary: Real atlases add thousands of sparse genes. Complete this evidence step: Repeat after a platform-scale shift. Write inputs, the mathematical operation, expected output, uncertainty and one observation that would contradict the claim. Repeat under a nearby defensible setting and compare feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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 labels, class means, E-test scores, selected genes, smoothing, likelihoods and validation. This connects information theory, probability and bioinformatics. The worked case is concrete: Students can begin with means, logarithms and proportions. Expose the boundary: A cell label does not establish mechanism or diagnosis. Complete this evidence step: Write a portable-classifier model 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 feature scores, weights, transformed coordinates, likelihoods, thresholds, mixtures, losses or probabilities. 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.