Combining cosmological data sets: hyperparameters and Bayesian evidence

A method is presented for performing joint analyses of cosmological data sets, in which the weight assigned to each data set is determined directly by its own statistical properties. The weights are considered in a Bayesian context as a set of hyperparameters, which are then marginalized over in order to recover the posterior distribution as a function only of the cosmological parameters of interest. In the case of a Gaussian likelihood function, this marginalization may be performed analytically. Calculation of the Bayesian evidence for the data, with and without the introduction of hyperparameters, enables a direct determination of whether the data warrant the introduction of weights into the analysis; this generalizes the standard likelihood ratio approach to model comparison. The method is illustrated by application to the classic toy problem of fitting a straight line to a set of data. A cosmological illustration of the technique is also presented, in which the latest measurements of the cosmic microwave background power spectrum are used to infer constraints on cosmological parameters.