Bayesian 'hyper-parameters' approach to joint estimation: the Hubble constant from CMB measurements

Recently several studies have jointly analysed data from different cosmolog ical probes with the motivation of estimating cosmological parameters. Here we generalize this procedure to allow freedom in the relative weights of v arious probes. This is done by including in the joint chi(2) function a set of 'hyper-parameters', which are dealt with using Bayesian considerations. The resulting algorithm, which assumes uniform priors on the log of the hy per-parameters, is very simple: instead of minimizing Sigma chi(j)(2) (wher e chi(j)(2) is per data set j) we propose to minimize sigma N-j ln(chi(j)(2 )) (where N-j is the number of data points per data set j). We illustrate t he method by estimating the Hubble constant H-0 from different sets of rece nt cosmic microwave background (CMB) experiments (including Saskatoon, Pyth on V, MSAM1, TOCO and Boomerang). The approach can be generalized for combi nations of cosmic probes, and for other priors on the hyper-parameters.