WHY MATHEMATICS?

Why Mathematics? | totalVI for RNA and Protein: Negative Binomial Counts, Background Mixtures, Latent Variables and Bayesian Testing

How totalVI combines RNA and surface-protein counts without pretending they have the same noise, then uses a joint latent state, mixture probabilities and posterior sampling for integration and differential analysis.

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

Model RNA overdispersion with a negative binomial law
Separate protein background and foreground with a conditional mixture
Learn a shared cell state through variational inference
Check posterior predictions, correlations and differential evidence

Chapter map

1. Two measurement tables
2. Shared cell state z
3. RNA library factor
4. Negative binomial RNA law
5. Protein background variable
6. Foreground–background mixture
7. Cell-specific mixture weight
8. Encoder factorization
9. Evidence lower bound
10. Reparameterized sampling
11. Batch-conditioned decoder
12. Normalized abundance estimate
13. Bayesian differential test
14. Worked protein mixture
15. Molecular correlation
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.

1. Two measurement tables

This step deserves a careful pause. For each cell, x stores gene counts and y stores antibody-derived protein counts, with batch labels s.

A concrete situation

One cell has 2,000 genes and 100 proteins.

What the mathematics does not license

Missing or mismatched panels need declared handling. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Publish dimensions and identifiers. 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. Shared cell state z

The useful question is not only how to calculate, but what the calculation assumes. An encoder maps both modalities into a posterior distribution for one low-dimensional biological state.

A concrete situation

RNA and protein jointly place a B cell in latent space.

What the mathematics does not license

Technical signals can leak into z. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Predict batches from the latent space. 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. RNA library factor

A small example makes the mechanism visible. A positive scaling variable models cell-specific RNA depth separately from normalized gene composition.

A concrete situation

Two similar cells differ in total UMIs.

What the mathematics does not license

Depth can correlate with biology. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Inspect factor by cell type and batch. 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. Negative binomial RNA law

This is where modelling choices begin to affect evidence. Gene counts follow a Gamma–Poisson mixture with mean l times rho and learned inverse dispersion theta.

A concrete situation

A gene mean of 4 can still produce zero or ten counts.

What the mathematics does not license

Distribution fit varies by gene and protocol. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Run posterior predictive 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.

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. Protein background variable

The central mathematical idea is concrete. A latent positive variable describes the local background level for each cell and protein.

A concrete situation

Ambient antibody signal raises CD marker counts in one batch.

What the mathematics does not license

Background is inferred, not observed. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use controls and replicates. 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. Foreground–background mixture

Here the geometry becomes operational. A Bernoulli indicator chooses between lower background and higher foreground negative binomial components.

A concrete situation

A count of eight receives a 0.7 foreground probability.

What the mathematics does not license

Two components may oversimplify complex staining. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Plot residual distributions. 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. Cell-specific mixture weight

This step deserves a careful pause. The background probability depends on latent state and batch rather than one global threshold per protein.

A concrete situation

The same count has different meaning in two cellular contexts.

What the mathematics does not license

Context dependence can encode circularity. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare with negative controls. 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. Encoder factorization

The useful question is not only how to calculate, but what the calculation assumes. Approximate posterior factors estimate distributions for cell state, RNA depth and protein background from observations.

A concrete situation

A cell receives means and variances for several hidden variables.

What the mathematics does not license

Mean-field assumptions can miss dependence. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Check posterior correlations. 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. Evidence lower bound

A small example makes the mechanism visible. Expected log likelihoods for RNA and protein are balanced against KL distances from priors.

A concrete situation

Improved reconstruction can increase posterior complexity.

What the mathematics does not license

One modality can dominate the objective. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report per-modality loss. 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. Reparameterized sampling

This is where modelling choices begin to affect evidence. Gaussian-like latent samples are written as mean plus scale times standard noise so gradients can pass through stochastic draws.

A concrete situation

Two epsilon draws produce nearby cell states.

What the mathematics does not license

Few samples increase Monte Carlo noise. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Repeat seeds and sample 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.

11. Batch-conditioned decoder

The central mathematical idea is concrete. Batch covariates enter decoders so technical shifts can be represented without forcing them into z.

A concrete situation

Two batches use different protein backgrounds.

What the mathematics does not license

Batch and disease can be confounded. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Design balanced 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. Normalized abundance estimate

Here the geometry becomes operational. Posterior decoding can estimate denoised gene frequency and protein foreground quantities at a chosen batch condition.

A concrete situation

A normalized CD4 value is compared across cells.

What the mathematics does not license

Normalized output is model-dependent. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Retain raw counts and settings. 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. Bayesian differential test

This step deserves a careful pause. Posterior samples estimate the probability that an effect exceeds a declared threshold between groups.

A concrete situation

P(log fold change greater than delta) equals 0.96.

What the mathematics does not license

Threshold and grouping choices affect evidence. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

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

14. Worked protein mixture

The useful question is not only how to calculate, but what the calculation assumes. Calculate two negative binomial component likelihoods, multiply by mixture weights and normalize to obtain a foreground probability.

A concrete situation

The posterior shifts from 0.4 to 0.85 when background likelihood falls.

What the mathematics does not license

Toy parameters hide uncertainty. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Vary dispersion and prior weight. 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. Molecular correlation

A small example makes the mechanism visible. Posterior samples support correlations between decoded genes and proteins while reducing some technical noise.

A concrete situation

RNA and surface protein show a positive association.

What the mathematics does not license

Correlation remains non-causal. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Validate with perturbation or time. 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 learning

This is where modelling choices begin to affect evidence. Version counts, panels, batches, priors, mixture, latent dimension and tests. This joins probability, statistics and multimodal AI.

A concrete situation

Students can start with Poisson mixtures.

What the mathematics does not license

A denoised value is an estimate. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Write a totalVI 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 totalVI audit treats RNA counts and surface-protein counts as two noisy views of one cell. A shared latent variable represents cellular state, while separate probabilistic decoders respect the measurement process of each modality. Gene counts follow a negative binomial distribution. Protein counts follow a two-component negative binomial mixture whose background probability and foreground scale depend on the cell and protein.

Record gene and antibody panels, raw counts, batch labels, library-size factors, latent dimension, priors, dispersion parameters, protein-background variables, encoder and decoder settings, evidence lower bound, training split and posterior sampling rule. The model separates biological state from technical factors mathematically; it does not make those factors observable without uncertainty.

Validate background separation with negative controls and known markers, compare RNA-only and protein-only baselines, leave entire batches outside tuning, inspect posterior predictive checks, and repeat differential-expression conclusions under plausible thresholds. A posterior mean is an estimate under a declared model, not a corrected measurement that replaces raw data.

Frequently asked questions

What is totalVI?

totalVI, Total Variational Inference, is a probabilistic model for paired RNA and surface-protein count data such as CITE-seq. It learns a joint representation while explicitly modelling technical factors.

Why not concatenate RNA and protein columns?

The modalities have different count ranges and noise. Protein measurements can contain ambient or nonspecific antibody background, while RNA counts are sparse and overdispersed. One generic distance can mix those effects.

How are RNA counts modelled?

The paper uses a negative binomial distribution whose mean combines a cell-specific RNA library-size factor with normalized gene frequencies predicted from latent cell state and batch.

How are protein counts modelled?

Each cell–protein count is generated by a negative binomial mixture with background and foreground components. A learned probability determines which component is more plausible.

What is a latent variable?

It is an unobserved numerical representation introduced by the model. In totalVI, the posterior mean of the cell-state variable supports visualization and clustering, but it is an estimate rather than a directly measured molecule.

Does totalVI prove differential expression?

No. It produces posterior evidence under its assumptions and comparison rule. Conclusions should include effect size, uncertainty, sensitivity analyses and external biological validation.

Primary source and fact-check note

Adam Gayoso, Zoë Steier, Romain Lopez et al., “Joint probabilistic modeling of single-cell multi-omic data with totalVI,” Nature Methods 18, 272–282 (2021), DOI 10.1038/s41592-020-01050-x. 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

Panel audit

Version every gene and antibody, clone, control, batch and missing-panel rule.

Count-model check

Use posterior predictive plots to test gene and protein count tails, zeros and dispersions.

Background validation

Compare inferred background with negative controls, empty droplets and known marker patterns.

Loss ledger

Report RNA likelihood, protein likelihood and every KL term separately.

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.

Latent leakage test

Measure how well batch, depth and panel can be predicted from the biological representation.

Differential sensitivity

Vary effect thresholds, priors, grouping and posterior samples.

External cohort

Keep a complete batch or study outside model and decision tuning.

Final claim

Name counts, negative binomial, mixture and posterior rule. Denoised abundance remains model-estimated.

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: Two measurement tables

Start with this article's mechanism: For each cell, x stores gene counts and y stores antibody-derived protein counts, with batch labels s. Work through the concrete case: One cell has 2,000 genes and 100 proteins. Make the boundary visible: Missing or mismatched panels need declared handling. Complete the evidence step: Publish dimensions and identifiers. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 2: Shared cell state z

Start with this article's mechanism: An encoder maps both modalities into a posterior distribution for one low-dimensional biological state. Work through the concrete case: RNA and protein jointly place a B cell in latent space. Make the boundary visible: Technical signals can leak into z. Complete the evidence step: Predict batches from the latent space. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 3: RNA library factor

Start with this article's mechanism: A positive scaling variable models cell-specific RNA depth separately from normalized gene composition. Work through the concrete case: Two similar cells differ in total UMIs. Make the boundary visible: Depth can correlate with biology. Complete the evidence step: Inspect factor by cell type and batch. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 4: Negative binomial RNA law

Start with this article's mechanism: Gene counts follow a Gamma–Poisson mixture with mean l times rho and learned inverse dispersion theta. Work through the concrete case: A gene mean of 4 can still produce zero or ten counts. Make the boundary visible: Distribution fit varies by gene and protocol. Complete the evidence step: Run posterior predictive checks. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

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: Protein background variable

Start with this article's mechanism: A latent positive variable describes the local background level for each cell and protein. Work through the concrete case: Ambient antibody signal raises CD marker counts in one batch. Make the boundary visible: Background is inferred, not observed. Complete the evidence step: Use controls and replicates. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 6: Foreground–background mixture

Start with this article's mechanism: A Bernoulli indicator chooses between lower background and higher foreground negative binomial components. Work through the concrete case: A count of eight receives a 0.7 foreground probability. Make the boundary visible: Two components may oversimplify complex staining. Complete the evidence step: Plot residual distributions. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 7: Cell-specific mixture weight

Start with this article's mechanism: The background probability depends on latent state and batch rather than one global threshold per protein. Work through the concrete case: The same count has different meaning in two cellular contexts. Make the boundary visible: Context dependence can encode circularity. Complete the evidence step: Compare with negative controls. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 8: Encoder factorization

Start with this article's mechanism: Approximate posterior factors estimate distributions for cell state, RNA depth and protein background from observations. Work through the concrete case: A cell receives means and variances for several hidden variables. Make the boundary visible: Mean-field assumptions can miss dependence. Complete the evidence step: Check posterior correlations. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

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: Evidence lower bound

Start with this article's mechanism: Expected log likelihoods for RNA and protein are balanced against KL distances from priors. Work through the concrete case: Improved reconstruction can increase posterior complexity. Make the boundary visible: One modality can dominate the objective. Complete the evidence step: Report per-modality loss. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 10: Reparameterized sampling

Start with this article's mechanism: Gaussian-like latent samples are written as mean plus scale times standard noise so gradients can pass through stochastic draws. Work through the concrete case: Two epsilon draws produce nearby cell states. Make the boundary visible: Few samples increase Monte Carlo noise. Complete the evidence step: Repeat seeds and sample counts. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 11: Batch-conditioned decoder

Start with this article's mechanism: Batch covariates enter decoders so technical shifts can be represented without forcing them into z. Work through the concrete case: Two batches use different protein backgrounds. Make the boundary visible: Batch and disease can be confounded. Complete the evidence step: Design balanced comparisons. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 12: Normalized abundance estimate

Start with this article's mechanism: Posterior decoding can estimate denoised gene frequency and protein foreground quantities at a chosen batch condition. Work through the concrete case: A normalized CD4 value is compared across cells. Make the boundary visible: Normalized output is model-dependent. Complete the evidence step: Retain raw counts and settings. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

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: Bayesian differential test

Start with this article's mechanism: Posterior samples estimate the probability that an effect exceeds a declared threshold between groups. Work through the concrete case: P(log fold change greater than delta) equals 0.96. Make the boundary visible: Threshold and grouping choices affect evidence. Complete the evidence step: Publish effect and sensitivity. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 14: Worked protein mixture

Start with this article's mechanism: Calculate two negative binomial component likelihoods, multiply by mixture weights and normalize to obtain a foreground probability. Work through the concrete case: The posterior shifts from 0.4 to 0.85 when background likelihood falls. Make the boundary visible: Toy parameters hide uncertainty. Complete the evidence step: Vary dispersion and prior weight. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 15: Molecular correlation

Start with this article's mechanism: Posterior samples support correlations between decoded genes and proteins while reducing some technical noise. Work through the concrete case: RNA and surface protein show a positive association. Make the boundary visible: Correlation remains non-causal. Complete the evidence step: Validate with perturbation or time. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 16: Governance and learning

Start with this article's mechanism: Version counts, panels, batches, priors, mixture, latent dimension and tests. This joins probability, statistics and multimodal AI. Work through the concrete case: Students can start with Poisson mixtures. Make the boundary visible: A denoised value is an estimate. Complete the evidence step: Write a totalVI model card. Write the input table, graph or count vector, the transformation, output dimensions, probability assumptions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare latent distributions, frozen gradients, mixture probabilities, symmetric divergences, graph edges, discriminator scores or posterior decisions as relevant. Use complete donors, studies, batches or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.