  Evidence synthesis models that combine multiple datasets of varying design,
to estimate quantities that cannot be directly observed, require the
formulation of complex probabilistic models that can be expressed as graphical
models. An assessment of whether the different datasets synthesised contribute
information that is consistent with each other, and in a Bayesian context, with
the prior distribution, is a crucial component of the model criticism process.
However, a systematic assessment of conflict suffers from the multiple testing
problem, through testing for conflict at multiple locations in a model. We
demonstrate the systematic use of conflict diagnostics, while accounting for
the multiple hypothesis tests of no conflict at each location in the graphical
model. The method is illustrated by a network meta-analysis to estimate
treatment effects in smoking cessation programs and an evidence synthesis to
estimate HIV prevalence in Poland.
