 Solution

Calling the effects exchangeable random variables means that their joint prior distribution is unchanged by permuting study labels. In a hierarchical Bayesian model, conditional independent draws ${\beta }_j\mid \mu ,\tau \sim N(\mu ,{\tau }^2)$ achieve this, and integrating shared hyperparameters induces dependence between studies. This permits partial pooling without asserting that all effects are identical.

The assumption is reasonable when the studies concern comparable treatments, populations, outcomes and follow-up, and no known study characteristic gives one effect a systematically different prior center. Relevant differences can instead enter a linear regression for study effects, after which the residual effects may be exchangeable. The numerical table counts deaths, so its event probabilities are mortality probabilities and ${\beta }_j<0$ indicates a lower mortality odds ratio. Interpreting those counts as beneficial responses would reverse the clinical meaning.