  When statistical analyses consider multiple data sources, Markov melding
provides a method for combining the source-specific Bayesian models. Markov
melding joins together submodels that have a common quantity. One challenge is
that the prior for this quantity can be implicit, and its prior density must be
estimated. We show that error in this density estimate makes the two-stage
Markov chain Monte Carlo sampler employed by Markov melding unstable and
unreliable. We propose a robust two-stage algorithm that estimates the required
prior marginal self-density ratios using weighted samples, dramatically
improving accuracy in the tails of the distribution. The stabilised version of
the algorithm is pragmatic and provides reliable inference. We demonstrate our
approach using an evidence synthesis for inferring HIV prevalence, and an
evidence synthesis of A/H1N1 influenza.
