= Solution
The overall mean $\mu$ and the <between-study heterogeneity> $\tau$ encode different information. A proper broad <normal distribution> prior such as $\mu\sim N(0,2^2)$ is one possible weak prior for the mean <log odds ratio>; information about the spread of trials alone does not determine its center.
A concrete <prior calibration for normal random-effect range> can make the factor-of-50 statement simultaneous across all six trials. Conditional on $\tau$, each pair difference has <normal distribution> $\beta_j-\beta_k\sim N(0,2\tau^2)$. Set $R=\log50$, $m=\binom62=15$, $z=\Phi^{-1}(1-0.05/(2m))$ and
$$
\boxed{\tau\sim\operatorname{Uniform}(0,A),\qquad A=\frac{R}{\sqrt2\,z}\simeq0.9424.}
$$
For every $\tau\le A$, each pair exceeds $R$ in absolute value with probability at most $0.05/m$. The <union bound> therefore gives $\mathbb P(\max_j\beta_j-\min_j\beta_j>R)\le0.05$, and integrating over the <uniform prior> preserves that bound. This is one explicit interpretation of “very unlikely”; a different elicited probability would change the bound. A smoother proper scale prior could be calibrated similarly.
The proposed $1/\tau$ <improper prior> is unsuitable. The observed-data <likelihood function> approaches the positive common-effect likelihood as $\tau\downarrow0$. After restricting $\mu$ and the intercepts to a compact interior region, it is bounded below there by a positive constant. Hence
$$
\int_0^\varepsilon L(\mu,\tau)\frac{d\tau}{\tau}=\infty.
$$
This is an <improper posterior from a log-uniform random-effect scale prior>. \b[Proper conditional sampling distributions do not repair the improper joint posterior], and an arbitrary tiny cutoff would make inference depend on that cutoff.
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