Group effects, uncertainty and shrinkage

Parametric Empirical Bayes

Parametric Empirical Bayes lets us ask whether model parameters differ across people, groups or symptoms, while respecting the uncertainty in each person's parameter estimates.

1. The problem

In Dynamic Causal Modelling, each participant has estimated parameters: for example synaptic gain, receptor time constants, connection strengths or delays. But these estimates are uncertain. PEB gives us a principled way to model how subject-level parameters depend on group-level predictors.

$\theta_i = X_i\beta + \varepsilon_i$

Here, $\theta_i$ is a subject-level parameter, $X_i$ contains group or symptom predictors, and $\beta$ contains the group-level effects.

2. Interactive PEB intuition

Each point is a subject-level parameter estimate. The vertical bars show uncertainty. The coloured dots show the shrinkage estimate: uncertain subjects are pulled more strongly toward the empirical group model.

Estimated group effect 0.00
Posterior probability 0.50
Interpretation PEB estimates the group effect while accounting for uncertainty in subject-level parameters.

3. Shrinkage is a feature, not a bug

Empirical Bayes uses the group-level model to provide empirical priors on subject-level parameters. Noisy estimates are not trusted blindly. Instead, they are partially pulled toward the group-informed expectation.

$\theta_i^{\text{PEB}} = w_i\theta_i^{\text{subject}} + (1-w_i)\theta_i^{\text{group}}$

The weight $w_i$ is larger when the subject-level estimate is precise and smaller when it is uncertain.

Subject level

Estimate each person's model parameters and uncertainty.

Group level

Explain parameters using predictors such as group, drug or symptoms.

Empirical priors

Use group structure to regularise individual estimates.

Posterior probability

Quantify confidence that an effect is positive or negative.

4. PEB as a hierarchy

PEB is easiest to understand as a hierarchy: data inform subject-level parameters, subject-level parameters inform group-level effects, and the group-level model feeds back as empirical priors.

Data
EEG, MEG, spectra
fit DCM
Subject parameters
with uncertainty
PEB
Group effects
β parameters
empirical priors

Why this matters for our lab

PEB is central when we want to connect mechanistic model parameters to clinical or experimental variables. For example, we might ask whether NMDA gain differs between groups, whether GABA time constants predict symptoms, or whether a drug changes synaptic parameters after administration. PEB lets us make these inferences while propagating uncertainty from the first-level DCMs.