WHY MATHEMATICS?

Why Mathematics? | BOOST-GP for Spatial Counts: Zero-Inflated Negative Binomials, Gaussian Processes, MCMC and Bayesian Gene Selection

How BOOST-GP separates extra zeros, overdispersion, sampling depth and spatial covariance before making Bayesian decisions about spatially variable genes.

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 observed counts and excess zeros directly
Separate sample depth from latent normalized expression
Compare spatial Gaussian-process and nonspatial covariance
Use posterior inclusion evidence with convergence and multiplicity checks

Chapter map

1. Observed counts
2. Extra-zero indicator
3. Zero probability
4. Negative-binomial mean
5. Overdispersion
6. Size factors
7. Log latent rate
8. Squared-exponential kernel
9. Length scale
10. Spatial inclusion variable
11. Hierarchical priors
12. Collapsed calculations
13. Gibbs and MH updates
14. Posterior inclusion probability
15. Bayes factor and BFDR
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. Observed counts

This step deserves a careful pause. Y_ij is a nonnegative integer for location i and gene j.

A concrete situation

One spot records eight reads.

What the mathematics does not license

Counts are not continuous Gaussian data. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Keep raw-count provenance. 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. Extra-zero indicator

The useful question is not only how to calculate, but what the calculation assumes. A latent Bernoulli variable distinguishes structural or false zeros from NB draws.

A concrete situation

A zero may come from the point mass.

What the mathematics does not license

Biological and technical zeros are not perfectly identifiable. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

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

3. Zero probability

A small example makes the mechanism visible. Each sample can have an extra-zero proportion.

A concrete situation

A low-depth spot has higher fitted pi.

What the mathematics does not license

Pi can absorb model misspecification. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Map it against depth. 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 mean

This is where modelling choices begin to affect evidence. The mean equals size factor times latent normalized expression.

A concrete situation

s_i times lambda_ij gives five.

What the mathematics does not license

Mean depends on factor normalization. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check identifiability. 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. Overdispersion

The central mathematical idea is concrete. Gene-specific dispersion lets variance exceed the mean.

A concrete situation

Variance equals mu plus mu squared over phi.

What the mathematics does not license

Tiny phi creates heavy variation. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Inspect posterior ranges. 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. Size factors

Here the geometry becomes operational. Total counts provide fixed sample scaling under a product constraint.

A concrete situation

A deep spot gets larger s_i.

What the mathematics does not license

Composition effects can bias total-count scaling. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Try alternatives. 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. Log latent rate

This step deserves a careful pause. Covariates plus a spatial random effect model log lambda.

A concrete situation

Cell type enters X beta.

What the mathematics does not license

Adjusting a covariate changes the question. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Pre-specify contrasts. 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. Squared-exponential kernel

The useful question is not only how to calculate, but what the calculation assumes. Correlation decays with squared coordinate distance.

A concrete situation

Two nearby spots have K_ij near one.

What the mathematics does not license

Abrupt boundaries violate smoothness. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Inspect residual maps. 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. Length scale

A small example makes the mechanism visible. l_j controls the correlation range for one gene.

A concrete situation

A large l_j gives broad smoothness.

What the mathematics does not license

Weak data create boundary estimates. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

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

10. Spatial inclusion variable

This is where modelling choices begin to affect evidence. Gamma_j chooses kernel K or identity I.

A concrete situation

One MCMC draw labels a gene spatial.

What the mathematics does not license

Selection depends on prior odds. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Run prior 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.

11. Hierarchical priors

The central mathematical idea is concrete. Beta, gamma, normal and inverse-gamma priors connect model components.

A concrete situation

Omega encodes expected spatial-gene fraction.

What the mathematics does not license

Vague priors can still be influential. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State hyperparameters. 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. Collapsed calculations

Here the geometry becomes operational. Integrating regression and scale parameters yields multivariate t forms.

A concrete situation

Sampling dimension is reduced.

What the mathematics does not license

Algebra relies on positive-definite matrices. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Check conditioning. 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. Gibbs and MH updates

This step deserves a careful pause. False-zero indicators use Gibbs; other parameters use Metropolis-Hastings.

A concrete situation

Four chains explore the posterior.

What the mathematics does not license

Poor mixing biases inclusion probabilities. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect chains and ESS. 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. Posterior inclusion probability

The useful question is not only how to calculate, but what the calculation assumes. PPI is the fraction of retained draws with gamma_j=1.

A concrete situation

Eight hundred of one thousand draws yield 0.8.

What the mathematics does not license

Autocorrelation lowers effective draws. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report Monte Carlo error. 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. Bayes factor and BFDR

A small example makes the mechanism visible. Posterior odds divided by prior odds quantify evidence; thresholds can control Bayesian FDR.

A concrete situation

A PPI cutoff targets five percent BFDR.

What the mathematics does not license

Bayesian and frequentist FDR differ. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Name the framework. 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, factors, priors, kernel, chains and threshold.

A concrete situation

A student computes one NB variance and kernel.

What the mathematics does not license

Posterior evidence is model-conditional. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Write a BOOST-GP audit 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

BOOST-GP models observed molecular counts without first turning them into continuous normalized values. A zero-inflated negative-binomial distribution separates an extra-zero process from overdispersed count sampling. Sample-specific size factors account for sequencing depth, while a latent normalized expression rate carries covariates and spatial structure.

On the log-rate scale, a gene can follow either a spatial Gaussian process with squared-exponential covariance or a nonspatial white-noise covariance. A binary inclusion variable chooses between these models. The gene-specific length scale controls how quickly spatial correlation decays, while covariates can adjust for known tissue or cell features.

Bayesian priors connect false-zero probabilities, dispersions, covariance scales, length scales and the expected proportion of spatial genes. Gibbs updates handle false-zero indicators and Metropolis-Hastings updates the remaining parameters. Posterior inclusion probability, Bayes factors and a Bayesian false-discovery-rate threshold summarize evidence.

A worked example can compare two locations with equal counts but different size factors, calculate one negative-binomial mean and construct a three-by-three squared-exponential kernel. Simulated inclusion draws then produce a posterior inclusion probability and posterior odds. This shows why count variation, missing detection and spatial covariance are distinct parts of the model.

Frequently asked questions

What is BOOST-GP?

BOOST-GP is a Bayesian hierarchical method that models spatial molecular counts with zero-inflated negative-binomial observations and Gaussian-process latent expression.

Why use a zero-inflated negative binomial?

The negative binomial allows variance above the mean, while an extra point mass at zero represents zeros beyond those expected from count sampling.

What does the size factor do?

A sample-specific multiplier accounts for global count-depth differences so latent expression is compared on a more meaningful scale.

How is a spatial gene represented?

Its log latent expression receives squared-exponential covariance based on coordinate distance; a nonspatial gene receives identity covariance.

How are genes selected?

MCMC samples a binary spatial-inclusion variable. Posterior inclusion probabilities produce posterior odds, Bayes factors and thresholds chosen for Bayesian FDR.

What is the practical cost?

MCMC and dense Gaussian-process covariance are computationally demanding and need multiple chains, convergence checks, prior sensitivity and careful reporting.

Primary source and fact-check note

Qiwei Li, Minzhe Zhang, Yang Xie and Guanghua Xiao, “Bayesian modeling of spatial molecular profiling data via Gaussian process,” Bioinformatics 37, 4129–4136 (2021), DOI 10.1093/bioinformatics/btab455. 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

Preserve raw counts and sample factors.

Zero audit

Map posterior extra-zero probabilities.

Dispersion audit

Inspect gene-specific NB variance.

Kernel audit

Check length-scale and boundary estimates.

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 audit

Vary expected spatial-gene proportion.

Chain audit

Use multiple chains, ESS and convergence.

Selection audit

Report PPI, Bayes factor and BFDR separately.

Final claim

Evidence is conditional on model and priors.

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: Observed counts

Begin with this mechanism: Y_ij is a nonnegative integer for location i and gene j. Work through the concrete case: One spot records eight reads. Make the limitation visible: Counts are not continuous Gaussian data. Complete the evidence step: Keep raw-count provenance. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 2: Extra-zero indicator

Begin with this mechanism: A latent Bernoulli variable distinguishes structural or false zeros from NB draws. Work through the concrete case: A zero may come from the point mass. Make the limitation visible: Biological and technical zeros are not perfectly identifiable. Complete the evidence step: Report posterior uncertainty. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 3: Zero probability

Begin with this mechanism: Each sample can have an extra-zero proportion. Work through the concrete case: A low-depth spot has higher fitted pi. Make the limitation visible: Pi can absorb model misspecification. Complete the evidence step: Map it against depth. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 4: Negative-binomial mean

Begin with this mechanism: The mean equals size factor times latent normalized expression. Work through the concrete case: s_i times lambda_ij gives five. Make the limitation visible: Mean depends on factor normalization. Complete the evidence step: Check identifiability. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

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: Overdispersion

Begin with this mechanism: Gene-specific dispersion lets variance exceed the mean. Work through the concrete case: Variance equals mu plus mu squared over phi. Make the limitation visible: Tiny phi creates heavy variation. Complete the evidence step: Inspect posterior ranges. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 6: Size factors

Begin with this mechanism: Total counts provide fixed sample scaling under a product constraint. Work through the concrete case: A deep spot gets larger s_i. Make the limitation visible: Composition effects can bias total-count scaling. Complete the evidence step: Try alternatives. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 7: Log latent rate

Begin with this mechanism: Covariates plus a spatial random effect model log lambda. Work through the concrete case: Cell type enters X beta. Make the limitation visible: Adjusting a covariate changes the question. Complete the evidence step: Pre-specify contrasts. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 8: Squared-exponential kernel

Begin with this mechanism: Correlation decays with squared coordinate distance. Work through the concrete case: Two nearby spots have K_ij near one. Make the limitation visible: Abrupt boundaries violate smoothness. Complete the evidence step: Inspect residual maps. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

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: Length scale

Begin with this mechanism: l_j controls the correlation range for one gene. Work through the concrete case: A large l_j gives broad smoothness. Make the limitation visible: Weak data create boundary estimates. Complete the evidence step: Report credible uncertainty. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 10: Spatial inclusion variable

Begin with this mechanism: Gamma_j chooses kernel K or identity I. Work through the concrete case: One MCMC draw labels a gene spatial. Make the limitation visible: Selection depends on prior odds. Complete the evidence step: Run prior sensitivity. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 11: Hierarchical priors

Begin with this mechanism: Beta, gamma, normal and inverse-gamma priors connect model components. Work through the concrete case: Omega encodes expected spatial-gene fraction. Make the limitation visible: Vague priors can still be influential. Complete the evidence step: State hyperparameters. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 12: Collapsed calculations

Begin with this mechanism: Integrating regression and scale parameters yields multivariate t forms. Work through the concrete case: Sampling dimension is reduced. Make the limitation visible: Algebra relies on positive-definite matrices. Complete the evidence step: Check conditioning. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

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: Gibbs and MH updates

Begin with this mechanism: False-zero indicators use Gibbs; other parameters use Metropolis-Hastings. Work through the concrete case: Four chains explore the posterior. Make the limitation visible: Poor mixing biases inclusion probabilities. Complete the evidence step: Inspect chains and ESS. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 14: Posterior inclusion probability

Begin with this mechanism: PPI is the fraction of retained draws with gamma_j=1. Work through the concrete case: Eight hundred of one thousand draws yield 0.8. Make the limitation visible: Autocorrelation lowers effective draws. Complete the evidence step: Report Monte Carlo error. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 15: Bayes factor and BFDR

Begin with this mechanism: Posterior odds divided by prior odds quantify evidence; thresholds can control Bayesian FDR. Work through the concrete case: A PPI cutoff targets five percent BFDR. Make the limitation visible: Bayesian and frequentist FDR differ. Complete the evidence step: Name the framework. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 16: Governance and learning

Begin with this mechanism: Version counts, factors, priors, kernel, chains and threshold. Work through the concrete case: A student computes one NB variance and kernel. Make the limitation visible: Posterior evidence is model-conditional. Complete the evidence step: Write a BOOST-GP audit card. Write the expression matrix, block labels, coordinates, graph edges, binary states, contingency cells, count means, dispersion, kernel entries, posterior odds, weight normalization or cross-products with named dimensions and units. Calculate one small example by hand, then vary one cluster number, start, threshold, edge, prior, length scale or neighbourhood range and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view, genes or ligand-receptor pairs outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.