Conservative cosmology: combining data with allowance for unknown
  systematics
  When combining data sets to perform parameter inference, the results will be
unreliable if there are unknown systematics in data or models. Here we
introduce a flexible methodology, BACCUS: BAyesian Conservative Constraints and
Unknown Systematics, which deals in a conservative way with the problem of data
combination, for any degree of tension between experiments. We introduce
hyperparameters that describe a bias in each model parameter for each class of
experiments. A conservative posterior for the model parameters is then obtained
by marginalization both over these unknown shifts and over the width of their
prior. We contrast this approach with an existing hyperparameter method in
which each individual likelihood is scaled, comparing the performance of each
approach and their combination in application to some idealized models. Using
only these rescaling hyperparameters is not a suitable approach for the current
observational situation, in which internal null tests of the errors are passed,
and yet different experiments prefer models that are in poor agreement. The
possible existence of large shift systematics cannot be constrained with a
small number of data sets, leading to extended tails on the conservative
posterior distributions. We illustrate our method with the case of the $H_0$
tension between results from the cosmic distance ladder and physical
measurements that rely on the standard cosmological model.
