WHY MATHEMATICS?

Why Mathematics? | BayesSpace Spatial Clustering: Student t Models, Potts Priors, MCMC and Subspot Resolution

How BayesSpace combines robust probability models and lattice neighbours, then estimates unobserved subspots without confusing model resolution with measurement resolution.

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 expression with a robust multivariate Student t distribution
Use a Potts prior to encourage local label agreement
Estimate labels and parameters through an MCMC chain
Split spots into constrained latent subspots and validate them

Chapter map

1. Principal-component input
2. Latent cluster label
3. Cluster mean vector
4. Shared precision matrix
5. Spot-specific weight
6. Student t marginal model
7. Lattice neighbours
8. Potts spatial prior
9. Cluster number q
10. Gibbs updates
11. Metropolis-Hastings labels
12. Burn-in and thinning
13. Subspot tessellation
14. Zero-sum jitter proposal
15. Acceptance-rate tuning
16. PC-to-gene mapping

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

1. Principal-component input

This is where modelling choices begin to affect evidence. Low-dimensional expression vectors reduce the modelled feature space.

A concrete situation

A spot keeps fifteen PCs.

What the mathematics does not license

PC choice can omit rare biology. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Repeat across dimensions. 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. Latent cluster label

The central mathematical idea is concrete. Each spot has an unobserved label from one to q.

A concrete situation

zi equals cluster 4.

What the mathematics does not license

Labels have no fixed biological meaning. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Annotate after inference. 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. Cluster mean vector

Here the geometry becomes operational. Every cluster has a centre in PC space.

A concrete situation

mu4 is a fifteen-value vector.

What the mathematics does not license

A single centre can hide substructure. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

4. Shared precision matrix

This step deserves a careful pause. One inverse covariance describes conditional variation across clusters.

A concrete situation

Lambda includes correlated PC residuals.

What the mathematics does not license

Shared precision may miss cluster-specific spread. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Compare model assumptions. 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. Spot-specific weight

The useful question is not only how to calculate, but what the calculation assumes. A latent scale reduces the impact of unusual observations.

A concrete situation

wi falls below one for an outlier.

What the mathematics does not license

Down-weighting can hide a real rare state. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Review low-weight spots. 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. Student t marginal model

A small example makes the mechanism visible. Gamma mixing of normal precision creates heavy tails.

A concrete situation

Degrees of freedom equal four.

What the mathematics does not license

Tail choice fixes robustness strength. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Sensitivity-test degrees of freedom. 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. Lattice neighbours

This is where modelling choices begin to affect evidence. Square or hexagonal assay geometry defines local adjacency.

A concrete situation

A Visium spot has up to six neighbours.

What the mathematics does not license

Missing tissue changes degree. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Plot the actual graph. 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. Potts spatial prior

The central mathematical idea is concrete. Equal neighbour labels receive a prior reward.

A concrete situation

Three matching neighbours increase support.

What the mathematics does not license

The prior can smooth across a true boundary. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Vary gamma. 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. Cluster number q

Here the geometry becomes operational. Biology or a pseudo-log-likelihood elbow selects the number of groups.

A concrete situation

q values from three to ten are compared.

What the mathematics does not license

An elbow is partly subjective. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

10. Gibbs updates

This step deserves a careful pause. Conditional distributions update means, precision and weights.

A concrete situation

One sweep revises every parameter block.

What the mathematics does not license

Conditional efficiency does not guarantee global mixing. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Run independent chains. 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. Metropolis-Hastings labels

The useful question is not only how to calculate, but what the calculation assumes. Proposed labels are accepted using likelihood and spatial prior ratios.

A concrete situation

A label move has acceptance probability 0.42.

What the mathematics does not license

Sticky chains can preserve initialization. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Check transition and occupancy traces. 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. Burn-in and thinning

A small example makes the mechanism visible. Early iterations are excluded and retained states may be spaced.

A concrete situation

The first 20 percent is discarded.

What the mathematics does not license

Discarding samples cannot rescue non-convergence. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use diagnostics before summaries. 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. Subspot tessellation

This is where modelling choices begin to affect evidence. Parent spots are divided into fixed child positions.

A concrete situation

Visium produces six subspots per spot.

What the mathematics does not license

Smaller polygons are not smaller measurements. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Label them latent estimates. 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. Zero-sum jitter proposal

The central mathematical idea is concrete. Child PC perturbations sum to zero and preserve the parent total.

A concrete situation

Six epsilon vectors add to zero.

What the mathematics does not license

Conservation does not identify the unique allocation. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Show posterior spread. 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. Acceptance-rate tuning

Here the geometry becomes operational. Jitter variance controls proposal size and MCMC movement.

A concrete situation

The target average acceptance is 25 to 40 percent.

What the mathematics does not license

A good rate does not prove a good model. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Inspect effective sample size. 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. PC-to-gene mapping

This step deserves a careful pause. Per-gene regression maps enhanced PCs back to log-expression space.

A concrete situation

XGBoost predicts one gene from enhanced PCs.

What the mathematics does not license

Regression adds a second model layer. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Validate genes separately. 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

BayesSpace treats a tissue map as a probability model rather than a coloured picture. Each spot has a low-dimensional expression vector, usually principal components, a latent cluster label and a lattice neighbourhood. Cluster centres and a shared precision matrix describe expression, while a Potts-model prior makes equal labels on adjacent spots more probable. The spatial preference is controlled by gamma; it is an assumption to test, not evidence that every true boundary is smooth.

The observation model uses a normal distribution conditional on a spot-specific weight. Integrating over gamma-distributed weights produces a multivariate Student t model, with the paper using four degrees of freedom. Outlying spots receive smaller weights and therefore less influence. Markov chain Monte Carlo updates cluster means, precision, weights and labels. Gibbs steps handle convenient conditional distributions, while Metropolis-Hastings steps accept or reject label proposals using expression fit and the spatial prior.

For enhanced resolution, each Visium spot is divided into six subspots and each older Spatial Transcriptomics spot into nine. Subspot principal-component values are latent because they were never measured. Proposals jitter the subspots while constraining their perturbations to sum to zero, so the parent spot total is preserved. The method then uses the same spatial clustering logic and maps enhanced PCs back to gene space with per-gene regression, by default XGBoost.

Subspot maps are estimates under the model. They do not become microscope observations merely because the pixels are smaller. A careful study checks chain mixing, burn-in, initialization, cluster number, gamma, degrees of freedom, jitter acceptance and agreement with held-out histology or markers. It also compares enhanced claims with what is supported at the original spot resolution.

Frequently asked questions

What is BayesSpace?

BayesSpace is a fully Bayesian method for spatial clustering and model-based subspot resolution enhancement of spatial transcriptomics.

Why use a Student t model?

Its heavier tails reduce the influence of outlying spots through latent observation weights.

What does the Potts prior do?

It increases prior support for a spot sharing a cluster label with its lattice neighbours, controlled by gamma.

How is the model fitted?

MCMC alternates Gibbs updates for convenient parameters with Metropolis-Hastings updates for labels and subspot proposals.

What is subspot resolution?

It is a latent reallocation of parent-spot expression into six Visium or nine older-ST subspots under spatial and conservation constraints.

Are subspot values measured?

No. They are posterior model estimates and require independent validation.

Primary source and fact-check note

Edward Zhao, Matthew R. Stone, Xing Ren et al., “Spatial transcriptomics at subspot resolution with BayesSpace,” Nature Biotechnology 39, 1375–1384 (2021), DOI 10.1038/s41587-021-00935-2. 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

Input audit

Version normalization, PCs and retained dimensions.

Prior audit

State q, gamma, mean and precision priors.

Chain audit

Keep starts, burn-in, thinning and convergence diagnostics.

Robustness audit

Review low spot weights rather than silently discarding them.

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.

Subspot audit

Preserve parent totals and report posterior spread.

Mapping audit

Validate every PC-to-gene regression separately.

Resolution audit

Compare claims at spot and subspot scales.

Final claim

Subspot expression is inferred, not directly measured.

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: Principal-component input

Begin with this mechanism: Low-dimensional expression vectors reduce the modelled feature space. Recreate the concrete case: A spot keeps fifteen PCs. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: PC choice can omit rare biology. Complete the transfer step: Repeat across dimensions. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 2: Latent cluster label

Begin with this mechanism: Each spot has an unobserved label from one to q. Recreate the concrete case: zi equals cluster 4. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Labels have no fixed biological meaning. Complete the transfer step: Annotate after inference. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 3: Cluster mean vector

Begin with this mechanism: Every cluster has a centre in PC space. Recreate the concrete case: mu4 is a fifteen-value vector. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: A single centre can hide substructure. Complete the transfer step: Inspect within-cluster residuals. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 4: Shared precision matrix

Begin with this mechanism: One inverse covariance describes conditional variation across clusters. Recreate the concrete case: Lambda includes correlated PC residuals. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Shared precision may miss cluster-specific spread. Complete the transfer step: Compare model assumptions. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

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

Workshop 5: Spot-specific weight

Begin with this mechanism: A latent scale reduces the impact of unusual observations. Recreate the concrete case: wi falls below one for an outlier. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Down-weighting can hide a real rare state. Complete the transfer step: Review low-weight spots. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 6: Student t marginal model

Begin with this mechanism: Gamma mixing of normal precision creates heavy tails. Recreate the concrete case: Degrees of freedom equal four. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Tail choice fixes robustness strength. Complete the transfer step: Sensitivity-test degrees of freedom. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 7: Lattice neighbours

Begin with this mechanism: Square or hexagonal assay geometry defines local adjacency. Recreate the concrete case: A Visium spot has up to six neighbours. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Missing tissue changes degree. Complete the transfer step: Plot the actual graph. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 8: Potts spatial prior

Begin with this mechanism: Equal neighbour labels receive a prior reward. Recreate the concrete case: Three matching neighbours increase support. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: The prior can smooth across a true boundary. Complete the transfer step: Vary gamma. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

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

Workshop 9: Cluster number q

Begin with this mechanism: Biology or a pseudo-log-likelihood elbow selects the number of groups. Recreate the concrete case: q values from three to ten are compared. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: An elbow is partly subjective. Complete the transfer step: Publish the full curve. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 10: Gibbs updates

Begin with this mechanism: Conditional distributions update means, precision and weights. Recreate the concrete case: One sweep revises every parameter block. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Conditional efficiency does not guarantee global mixing. Complete the transfer step: Run independent chains. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 11: Metropolis-Hastings labels

Begin with this mechanism: Proposed labels are accepted using likelihood and spatial prior ratios. Recreate the concrete case: A label move has acceptance probability 0.42. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Sticky chains can preserve initialization. Complete the transfer step: Check transition and occupancy traces. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 12: Burn-in and thinning

Begin with this mechanism: Early iterations are excluded and retained states may be spaced. Recreate the concrete case: The first 20 percent is discarded. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Discarding samples cannot rescue non-convergence. Complete the transfer step: Use diagnostics before summaries. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

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

Workshop 13: Subspot tessellation

Begin with this mechanism: Parent spots are divided into fixed child positions. Recreate the concrete case: Visium produces six subspots per spot. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Smaller polygons are not smaller measurements. Complete the transfer step: Label them latent estimates. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 14: Zero-sum jitter proposal

Begin with this mechanism: Child PC perturbations sum to zero and preserve the parent total. Recreate the concrete case: Six epsilon vectors add to zero. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Conservation does not identify the unique allocation. Complete the transfer step: Show posterior spread. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 15: Acceptance-rate tuning

Begin with this mechanism: Jitter variance controls proposal size and MCMC movement. Recreate the concrete case: The target average acceptance is 25 to 40 percent. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: A good rate does not prove a good model. Complete the transfer step: Inspect effective sample size. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.

Workshop 16: PC-to-gene mapping

Begin with this mechanism: Per-gene regression maps enhanced PCs back to log-expression space. Recreate the concrete case: XGBoost predicts one gene from enhanced PCs. Write PC values, latent weights, cluster means, precision entries, neighbour indicators, Potts rewards, likelihood ratios, acceptance probabilities, chain traces, subspot jitters, posterior frequencies and gene-regression residuals with direction, dimensions, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Regression adds a second model layer. Complete the transfer step: Validate genes separately. Change exactly one gene filter, PC dimension, cluster count, prior, gamma, degrees of freedom, lattice edge, initialization, chain length, burn-in, jitter scale, subspot layout, radius, graph degree, attention score, alpha mixture, latent width, tied parameter, mask rate, mask pattern, KL weight, cluster target, corruption rule, negative sample, pairwise distance normalization, Leiden resolution, pseudotime root, tissue section or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, graph, prior, objective, optimization, posterior or embedding, clustering, biological interpretation and independent validation on separate notebook lines. Reserve whole donors, slides, tissues, patients or platforms outside tuning when testing transfer.