WHY MATHEMATICS?

Why Mathematics? | Spacia Multiple-Sender Communication: Bayesian Multiple-Instance Learning, Spatial Distance and Variable Selection

How Spacia treats each receiver as a bag of possible senders, uses spatial distance to guide latent sender selection and quantifies conditional sender-to-receiver associations with Bayesian uncertainty.

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

Translate receiver neighbourhoods into bags and instances
Use distance as prior evidence rather than causal proof
Estimate joint multi-sender effects with Bayesian priors
Report posterior uncertainty and donor-level validation

Chapter map

1. Receiver bag
2. Sender instances
3. Spatial candidate radius
4. Distance feature
5. Latent influence indicator
6. Monotone distance effect
7. Sender molecular features
8. Receiver outcome
9. Tier-one sender selection
10. Tier-two regulatory coefficient
11. Bayesian prior
12. Posterior inference
13. Joint multi-sender effect
14. Variable selection
15. Posterior predictive check
16. Transfer and causal limits

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

1. Receiver bag

Here the geometry becomes operational. One receiver cell defines one prediction unit.

A concrete situation

Receiver r is one bag.

What the mathematics does not license

Receiver quality affects every association. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

2. Sender instances

This step deserves a careful pause. Nearby candidate senders are instances inside the bag.

A concrete situation

Bag r contains eight cells.

What the mathematics does not license

Candidate sets may omit distant endocrine signals. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

State the biological scope. 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. Spatial candidate radius

The useful question is not only how to calculate, but what the calculation assumes. Coordinates and a rule select possible senders.

A concrete situation

All cells within 100 micrometres enter.

What the mathematics does not license

Radius changes bag size and confounding. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Run a radius sensitivity 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.

4. Distance feature

A small example makes the mechanism visible. Each sender-receiver separation becomes a numeric predictor.

A concrete situation

d-ir equals 35 micrometres.

What the mathematics does not license

Euclidean distance ignores barriers. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare morphology-aware distance. 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. Latent influence indicator

This is where modelling choices begin to affect evidence. Delta-ir marks modelled sender contribution.

A concrete situation

Posterior Pr(delta=1) is 0.82.

What the mathematics does not license

It is model-based, not observed. The same reasoning helps students move from an exercise to a real dataset.

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.

6. Monotone distance effect

The central mathematical idea is concrete. The prior influence changes in one direction with distance.

A concrete situation

Closer senders receive higher prior support.

What the mathematics does not license

True biology may be non-monotone. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Test flexible 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. Sender molecular features

Here the geometry becomes operational. Genes or pathways describe each candidate sender.

A concrete situation

Sender i has pathway score 1.4.

What the mathematics does not license

Correlated genes impair identifiability. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Group or regularize features. 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. Receiver outcome

This step deserves a careful pause. A receiver gene or pathway is the bag label.

A concrete situation

Receiver r expresses target T at six.

What the mathematics does not license

Counts need an appropriate likelihood. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Check residuals and dispersion. 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. Tier-one sender selection

The useful question is not only how to calculate, but what the calculation assumes. The first model estimates which instances influence a bag.

A concrete situation

Two of eight senders have high delta probability.

What the mathematics does not license

Spatial confounders can drive selection. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Add tissue and type covariates. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

10. Tier-two regulatory coefficient

A small example makes the mechanism visible. Selected sender features predict receiver outcomes.

A concrete situation

Beta-gk equals 0.45.

What the mathematics does not license

A coefficient is conditional association. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

11. Bayesian prior

This is where modelling choices begin to affect evidence. Priors control effect size and sparsity.

A concrete situation

Beta follows a zero-centred shrinkage prior.

What the mathematics does not license

Strong priors can dominate small data. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Publish prior 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.

12. Posterior inference

The central mathematical idea is concrete. Data and priors produce distributions over unknowns.

A concrete situation

A 95% credible interval is 0.1 to 0.8.

What the mathematics does not license

Approximate sampling may not converge. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Report diagnostics and effective 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.

13. Joint multi-sender effect

Here the geometry becomes operational. Several instances contribute simultaneously.

A concrete situation

Three stromal senders jointly explain T.

What the mathematics does not license

Collinearity can split attribution. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

14. Variable selection

This step deserves a careful pause. Posterior inclusion ranks sender features.

A concrete situation

Gene G has inclusion probability 0.9.

What the mathematics does not license

Selection after screening can be optimistic. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

15. Posterior predictive check

The useful question is not only how to calculate, but what the calculation assumes. Simulated receiver outcomes are compared with observed data.

A concrete situation

Observed T lies inside replicated distribution.

What the mathematics does not license

Fit does not identify mechanism uniquely. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Check structured 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.

16. Transfer and causal limits

A small example makes the mechanism visible. Whole donors and perturbations test generalization.

A concrete situation

A held-out tissue preserves the association.

What the mathematics does not license

Cross-sectional expression cannot establish direction in time. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Seek perturbation and protein evidence. 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

Spacia frames cell-cell communication as Bayesian multiple-instance learning. Each receiver cell is a bag, and nearby candidate sender cells are the instances inside that bag. Receiver gene or pathway expression is the bag-level outcome. Sender molecular features and spatial separation are used to learn which instances plausibly influence that receiver and which sender variables are associated with the receiver outcome.

The first tier contains a latent indicator for whether each candidate sender contributes to the receiver. Spatial distance informs this selection through a monotone relationship: nearer candidates can receive greater prior support, while the data may still down-weight them. The distance rule encodes opportunity and must not be confused with a measured causal connection. Radius choice, coordinate quality and tissue geometry affect the candidate set.

The second tier estimates how sender genes or pathways relate to receiver genes or pathways among selected sender-receiver relationships. Bayesian priors and posterior inference express uncertainty, while variable selection helps avoid treating every sender feature as relevant. Because several senders may influence one receiver simultaneously, the coefficients are conditional on the other included senders and features rather than isolated pairwise correlations.

Posterior probabilities and credible intervals support ranked hypotheses, but their calibration depends on the likelihood, priors and sampling scheme. Donor-level validation, posterior predictive checks and sensitivity analyses are essential. Spacia is valuable because its mathematics matches a real biological ambiguity: a receiver can sit near many possible senders, and the model must reason about joint influence without pretending that proximity alone identifies the source.

Frequently asked questions

What is Spacia?

Spacia is a Bayesian multiple-instance learning framework for mapping sender-to-receiver relationships in spatially resolved transcriptomics.

What is a bag in Spacia?

A receiver cell is the bag, and nearby candidate sender cells are its instances.

How does distance enter the model?

Spatial closeness informs a monotone prior or selection relationship for latent sender influence.

What does the latent sender indicator mean?

It represents the modelled probability that a candidate sender contributes to the receiver under the specified data and priors.

Why model several senders jointly?

A receiver can be influenced by multiple nearby cells, and joint modelling separates conditional contributions better than isolated pairwise scores.

Does a high posterior probability establish causality?

No. It quantifies uncertainty within the model; perturbations, temporal evidence and independent samples remain necessary.

Primary source and fact-check note

James Zhu, Yunguan Wang, Woo Yong Chang et al., “Mapping cellular interactions from spatially resolved transcriptomics data,” Nature Methods 21, 1830–1842 (2024), DOI 10.1038/s41592-024-02408-1. 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

Bag audit

Publish receiver definition and every candidate sender.

Distance audit

State units, radius, metric and monotonicity.

Indicator audit

Report posterior delta probabilities, not binary truth.

Prior audit

Use and publish prior predictive checks.

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.

Coefficient audit

Name conditional associations and collinearity.

Inference audit

Report convergence and effective sample size.

Validation audit

Hold out whole donors and use posterior predictive checks.

Final claim

Bayesian confidence is conditional on model and data, not causal proof.

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: Receiver bag

Begin with this mechanism: One receiver cell defines one prediction unit. Recreate the concrete case: Receiver r is one bag. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Receiver quality affects every association. Complete the transfer step: Filter and record receiver covariates. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.

Workshop 2: Sender instances

Begin with this mechanism: Nearby candidate senders are instances inside the bag. Recreate the concrete case: Bag r contains eight cells. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Candidate sets may omit distant endocrine signals. Complete the transfer step: State the biological scope. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.

Workshop 3: Spatial candidate radius

Begin with this mechanism: Coordinates and a rule select possible senders. Recreate the concrete case: All cells within 100 micrometres enter. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Radius changes bag size and confounding. Complete the transfer step: Run a radius sensitivity curve. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.

Workshop 4: Distance feature

Begin with this mechanism: Each sender-receiver separation becomes a numeric predictor. Recreate the concrete case: d-ir equals 35 micrometres. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Euclidean distance ignores barriers. Complete the transfer step: Compare morphology-aware distance. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions 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: Latent influence indicator

Begin with this mechanism: Delta-ir marks modelled sender contribution. Recreate the concrete case: Posterior Pr(delta=1) is 0.82. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: It is model-based, not observed. Complete the transfer step: Report posterior uncertainty. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.

Workshop 6: Monotone distance effect

Begin with this mechanism: The prior influence changes in one direction with distance. Recreate the concrete case: Closer senders receive higher prior support. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: True biology may be non-monotone. Complete the transfer step: Test flexible alternatives. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.

Workshop 7: Sender molecular features

Begin with this mechanism: Genes or pathways describe each candidate sender. Recreate the concrete case: Sender i has pathway score 1.4. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Correlated genes impair identifiability. Complete the transfer step: Group or regularize features. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.

Workshop 8: Receiver outcome

Begin with this mechanism: A receiver gene or pathway is the bag label. Recreate the concrete case: Receiver r expresses target T at six. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Counts need an appropriate likelihood. Complete the transfer step: Check residuals and dispersion. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions 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: Tier-one sender selection

Begin with this mechanism: The first model estimates which instances influence a bag. Recreate the concrete case: Two of eight senders have high delta probability. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Spatial confounders can drive selection. Complete the transfer step: Add tissue and type covariates. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.

Workshop 10: Tier-two regulatory coefficient

Begin with this mechanism: Selected sender features predict receiver outcomes. Recreate the concrete case: Beta-gk equals 0.45. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: A coefficient is conditional association. Complete the transfer step: Avoid regulatory-causation language. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.

Workshop 11: Bayesian prior

Begin with this mechanism: Priors control effect size and sparsity. Recreate the concrete case: Beta follows a zero-centred shrinkage prior. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Strong priors can dominate small data. Complete the transfer step: Publish prior predictive checks. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.

Workshop 12: Posterior inference

Begin with this mechanism: Data and priors produce distributions over unknowns. Recreate the concrete case: A 95% credible interval is 0.1 to 0.8. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Approximate sampling may not converge. Complete the transfer step: Report diagnostics and effective size. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions 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: Joint multi-sender effect

Begin with this mechanism: Several instances contribute simultaneously. Recreate the concrete case: Three stromal senders jointly explain T. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Collinearity can split attribution. Complete the transfer step: Test sender-group stability. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.

Workshop 14: Variable selection

Begin with this mechanism: Posterior inclusion ranks sender features. Recreate the concrete case: Gene G has inclusion probability 0.9. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Selection after screening can be optimistic. Complete the transfer step: Use held-out validation. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.

Workshop 15: Posterior predictive check

Begin with this mechanism: Simulated receiver outcomes are compared with observed data. Recreate the concrete case: Observed T lies inside replicated distribution. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Fit does not identify mechanism uniquely. Complete the transfer step: Check structured residuals. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.

Workshop 16: Transfer and causal limits

Begin with this mechanism: Whole donors and perturbations test generalization. Recreate the concrete case: A held-out tissue preserves the association. Write bag sizes, sender instances, distances, latent indicators, pathway scores, receiver outcomes, prior scales, coefficients, inclusion probabilities and predictive residuals with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Cross-sectional expression cannot establish direction in time. Complete the transfer step: Seek perturbation and protein evidence. Change exactly one catalogue, normalization, threshold, neighbourhood radius, distance kernel, graph layer, hidden dimension, attention seed, corruption rule, latent prior, decoder mask, KL weight, sender bag, spatial metric, shrinkage prior or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, graph construction, probabilistic assumption, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, tissues or conditions outside tuning when testing transfer.