  Meta-analyses of clinical trials targeting rare events face particular
challenges when the data lack adequate numbers of events for all treatment
arms. Especially when the number of studies is low, standard meta-analysis
methods can lead to serious distortions because of such data sparsity. To
overcome this, we suggest the use of weakly informative priors (WIP) for the
treatment effect parameter of a Bayesian meta-analysis model, which may also be
seen as a form of penalization. As a data model, we use a binomial-normal
hierarchical model (BNHM) which does not require continuity corrections in case
of zero counts in one or both arms. We suggest a normal prior for the log odds
ratio with mean 0 and standard deviation 2.82, which is motivated (1) as a
symmetric prior centred around unity and constraining the odds ratio to within
a range from 1/250 to 250 with 95 % probability, and (2) as consistent with
empirically observed effect estimates from a set of $\mbox{$37\,773$}$
meta-analyses from the Cochrane Database of Systematic Reviews. In a simulation
study with rare events and few studies, our BNHM with a WIP outperformed a
Bayesian method without a WIP and a maximum likelihood estimator in terms of
smaller bias and shorter interval estimates with similar coverage. Furthermore,
the methods are illustrated by a systematic review in immunosuppression of rare
safety events following paediatric transplantation. A publicly available
$\textbf{R}$ package, $\texttt{MetaStan}$, is developed to automate the
$\textbf{Stan}$ implementation of meta-analysis models using WIPs.
