Spike-and-Slab Commensurability Priors

Why Put a Spike on Commensurability?

A commensurate prior lets current-trial information borrow from a historical treatment effect, with the amount of borrowing controlled by a precision parameter. A finite precision allows the effects to differ: larger precision concentrates the current effect more tightly around the historical effect. But a prior that only assigns finite precision does not give positive probability to exact equality. A spike-and-slab prior addresses this by assigning one part of the prior to exact commensurability and the other to a distribution over finite precisions.

This distinction is clinically useful. The spike represents a model in which the underlying historical and current effects are equal. The slab represents models in which they may differ, with the precision determining how strongly the current effect is expected to remain close to the historical effect. Neither component says that a historical estimate is known without uncertainty.

For calculations, it is useful to represent the two components with an indicator \(Z\). Set \(Z=1\) for the spike and \(Z=0\) for the slab, with \(P(Z=1)=w\). When \(Z=1\), \(\theta_C=\theta_H\) exactly; when \(Z=0\), draw a finite \(\kappa\) from \(G\) and use the normal prior above. A continuous slab places probability zero on any one exact finite value of \(\kappa\), while the spike is an atom with positive prior probability.

Model and Posterior Component Probabilities

Suppose historical data \(D_H\) have been analyzed to give \(\theta_H\mid D_H\sim N(m_H,V_H)\). Let \(Y\) be a current-trial estimate of the treatment effect. Use the sampling model \(Y\mid\theta_C\sim N(\theta_C,s^2)\), where \(s\) is the known standard error (or a justified normal-approximation standard error). Assume the current and historical data are conditionally independent given their respective effects, and that the prior and sampling models are appropriate for the endpoint and trial design.

Under the spike, integrating over the historical uncertainty gives \(Y\mid Z=1,D_H\sim N(m_H,s^2+V_H)\). Under a slab value \(\kappa\), the prior distribution of \(\theta_C\) after integrating over \(\theta_H\mid D_H\) is \(N(m_H,V_H+1/\kappa)\). Thus \(Y\mid Z=0,\kappa,D_H\) is normal with mean \(m_H\) and variance \(s^2+V_H+1/\kappa\). These distributions provide the component marginal likelihoods: the probability densities assigned to the observed current estimate by each component.

Here \(\phi(y;\mu,v)\) is the normal density at \(y\) with mean \(\mu\) and variance \(v\). The subscript 1 denotes the spike and 0 the slab; \(m_0(y)\) averages the slab likelihood over its prior distribution \(G\) on \(\kappa\). Bayes' rule gives the posterior probability of exact commensurability:

Conditional on being in the slab, the posterior distribution of \(\kappa\) is proportional to its prior distribution times its component likelihood. This also gives a precise way to report approximate borrowing. For a prespecified threshold \(\kappa_0\), the posterior probability of slab borrowing at least that precise is:

The threshold must be chosen and interpreted on the effect scale: large \(\kappa\) means small prior variance \(1/\kappa\), hence stronger approximate borrowing. A threshold probability is not the posterior probability of exact equality. The latter is the spike probability alone.

Worked Posterior Calculations

Justification and Practical Interpretation

The marginal distributions and component updates follow from the sum of independent normal variables. Under the slab, conditional on \(D_H\), write \(\theta_C=\theta_H+\epsilon\), where \(\theta_H\sim N(m_H,V_H)\) and \(\epsilon\sim N(0,1/\kappa)\) independently. Then \(\theta_C\sim N(m_H,V_H+1/\kappa)\). Adding the sampling error in \(Y=\theta_C+\eta\), with \(\eta\sim N(0,s^2)\) independent, gives \(Y\sim N(m_H,V_H+1/\kappa+s^2)\). Under the spike, \(\epsilon=0\), so the same argument gives variance \(V_H+s^2\). Applying Bayes' rule to the two component probabilities yields the posterior formula above; applying it to a slab interval yields the threshold-probability formula.

A posterior probability is conditional on the model, its prior mixture weight, and the observed data. A low spike probability can reflect conflict with the exact-equality model, but it does not by itself identify why the historical and current evidence differ. Trial populations, endpoint measurement, treatment delivery, and chance variation may all matter. Likewise, a high spike probability is not a guarantee that borrowing is scientifically appropriate under every unmodeled source of difference.

Diagnostics, Alternatives, and Limitations

Check that the normal approximation for the current estimate is credible, that the standard error is on the same treatment-effect scale as the historical posterior, and that historical data have not been counted twice. Inspect predictive distributions under each component: if the observed current estimate lies in a tail under both, the model may fit poorly even if one component receives most of the posterior probability. Recalculate posterior probabilities over plausible values of \(w\) and slab distributions \(G\); a conclusion that changes sharply under modest changes is prior-sensitive.

A fixed finite commensurability precision is simpler, but it cannot assign positive probability to exact equality or adapt across precision values. A spike-and-slab prior explicitly separates exact equality from finite differences, while a robust mixture that instead mixes prior distributions for the current effect represents a different modeling choice. The spike-and-slab approach has interpretable component probabilities, but those probabilities depend on the specified component distributions and prior weight. The threshold \(\kappa_0\) for approximate borrowing is also a reporting choice, not a universal scientific boundary.

A practical failure mode is to interpret \(\kappa\) as a direct measure of borrowing without accounting for uncertainty in the historical effect. The actual marginal prior variance around \(m_H\) under the slab is \(V_H+1/\kappa\), not merely \(1/\kappa\). Even as \(\kappa\) grows very large, uncertainty \(V_H\) remains unless historical information is effectively known without uncertainty. In addition, a posterior spike probability is not a posterior estimate of the treatment effect and should be reported alongside the component-conditional or mixture posterior for \(\theta_C\).