Bayesian analyses require that all variable model parameters are given a prior probability distribution. This can pose a challenge for analyses where multiple experiments are combined if these experiments use different parametrisations for their nuisance parameters. If the parameters in the two models describe exactly the same physics, they should be 100% correlated in the prior. If the parameters describe independent physics, they should be uncorrelated. But if they describe related or overlapping physics, it is not trivial to determine what the joint prior distribution should look like. Even if the priors for each experiment are well motivated, the unknown correlations between them can have unintended consequences for the posterior probability of the parameters of interest, potentially leading to underestimated uncertainties. In this paper we show that it is possible to choose a prior parametrisation that ensures conservative posterior uncertainties for the parameters of interest under some very general assumptions.

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