= Prior calibration for normal random-effect range
{title2=$\tau\le R/(\sqrt2\,z_{1-\varepsilon/(2\binom J2)})$}
For $J\ge2$ conditionally independent effects $\beta_j\sim N(\mu,\tau^2)$, each pair difference has <normal distribution> $N(0,2\tau^2)$. With $m=\binom J2$, an upper bound $A=R/[\sqrt2\,\Phi^{-1}(1-\varepsilon/(2m))]$ on $\tau$ ensures, by the <union bound>, that $\mathbb P(\max_j\beta_j-\min_j\beta_j>R)\le\varepsilon$. Any proper scale <prior distribution> supported on $(0,A)$ preserves this bound after averaging. This is conservative simultaneous calibration, rather than the weaker statement about one selected pair. For <log odds ratios>, a bound $R$ corresponds to a ratio-of-odds-ratios bound $e^R$.
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