Learning multivariate effect patterns for MASH

This vignette explains how to estimate a mixture-of-covariances prior, fit MASH and interpret the learned sharing patterns.

Learning goals

After completing this vignette, you will be able to explain what the mixture prior represents, distinguish prior estimation from posterior fitting, run the three-step example, and interpret covariance components without treating them as discrete biological clusters.

Background and method

MASH models effects across many related conditions using a mixture of multivariate normal distributions. Each covariance component represents a possible pattern of effect sharing: condition-specific, broadly shared, correlated among a subset of conditions or learned from the data. Mixture weights describe how strongly the observed effects support these patterns.

Prior estimation and posterior inference are separate operations. mixture_prior.ipynb learns data-driven covariance components from strong signals and combines them with canonical components. mash_fit.ipynb fits the mixture to the full analysis set and computes posterior effect estimates, uncertainties and measures of sharing. Reusing a prior is appropriate only when its conditions and effect architecture are compatible with the target analysis.

The example uses small synthetic 34-context inputs. It demonstrates object flow and interpretation, not a biological reference prior.

Worked example

The analysis has three sequential steps: estimate the prior, fit MASH, then visualize the covariance components. The paths below define the intended protocol-example bundle, but the complete bundle is not currently shipped: only the prior-estimation fixture is available. The commands show the required interface until the missing MASH input and residual-correlation files are added.

Role	Example path
Prior-estimation input	input/mash/protocol_example.mashr_input.rds
MASH input	input/mash/protocol_example.EE.mash.rds
Residual correlation matrix	input/mash/protocol_example.EE.V_simple.rds

The matrices must use the same condition order and compatible units. The prior-estimation object contains random, strong and null subsets plus XtX; the MASH input contains the effects and standard errors to which the learned prior is applied.

Step 1. Estimate the mixture prior

ed_bovy estimates data-driven covariance components and assembles them with canonical components.

Timing: TBD

sos run pipeline/mixture_prior.ipynb ed_bovy \
  --output-prefix protocol_example \
  --data input/mash/protocol_example.mashr_input.rds \
  --cwd output/mixture_prior

Step 2. Fit MASH

mash combines the input effects, residual correlation and estimated prior to compute posterior summaries.

Timing: TBD

sos run pipeline/mash_fit.ipynb mash \
  --output-prefix protocol_example_mash \
  --data input/mash/protocol_example.EE.mash.rds \
  --vhat-data input/mash/protocol_example.EE.V_simple.rds \
  --prior-data output/mixture_prior/protocol_example.EE.prior.rds \
  --effect-model EE \
  --compute-posterior \
  --cwd output/mash_fit

Step 3. Visualize the learned covariance components

plot_U draws heatmaps of the covariance matrices in the fitted prior.

Timing: TBD

sos run pipeline/mixture_prior.ipynb plot_U \
  --output-prefix protocol_example_plots \
  --data output/mixture_prior/protocol_example.EE.prior.rds \
  --cwd output/mixture_prior

The chain produces:

• output/mixture_prior/protocol_example.EE.prior.rds — data-driven and canonical covariance components with their mixture structure.

• output/mash_fit/protocol_example_mash.*.rds — fitted MASH model and, when requested, posterior summaries.

• output/mixture_prior/protocol_example_plots.EE.prior.pdf — covariance-component heatmaps.

Run the steps in order. Step 2 uses the prior from Step 1; Step 3 visualizes that same prior.

Command reference

sos run pipeline/mixture_prior.ipynb -h
sos run pipeline/mash_fit.ipynb -h

Results and interpretation

Mixture-prior result

The prior contains covariance matrices and mixture information. Inspect dimensions, names and weights rather than printing every matrix.

Result	Interpretation
Covariance component	Candidate pattern of effect sharing across conditions
Diagonal values	Relative effect variance within conditions
Off-diagonal values	Direction and strength of shared effects
Mixture weight	Support for a component in the fitted effect distribution
Posterior summaries	Shrunk effects and uncertainty after combining data with the mixture

prior <- readRDS("output/mixture_prior/protocol_example.EE.prior.rds")
cat("Class:", class(prior), "\n")
cat("Top-level fields:", paste(names(prior), collapse = ", "), "\n")

fit_files <- list.files("output/mash_fit", pattern = "^protocol_example_mash.*\\.rds$", full.names = TRUE)
fit <- readRDS(fit_files[[1]])
cat("MASH result fields:", paste(names(fit), collapse = ", "), "\n")

A covariance heatmap is a pattern used by the mixture model, not a cluster assignment for samples or conditions. Strong positive off-diagonal structure supports effects with similar direction across conditions; near-diagonal structure supports condition-specific effects. Interpret a pattern together with its estimated weight and posterior results. Components can overlap, and several components may approximate the same broad biological pattern.

Limitations and common pitfalls

• Condition names and ordering must match across all matrices.

• Strong-effect subsets used to learn covariances can bias the prior if they are unrepresentative.

• A prior learned in one study should not be transferred automatically to another tissue set or measurement scale.

• Residual correlation V and biological effect covariance are different quantities.

• Similar covariance components can be non-identifiable; interpret broad patterns rather than every component literally.

• The complete protocol-example input bundle is currently missing, so the three-step worked example cannot yet run end to end.

• Synthetic inputs demonstrate mechanics, not expected biological sharing.

Next steps

Use the fitted posterior summaries to identify effects that are shared or condition-specific, then compare those conclusions with sensitivity analyses using alternative covariance sets or residual-correlation estimates. Record the condition order and prior provenance with every downstream result.