  We introduce a statistical measure of the effective model complexity, called
the Bayesian complexity. We demonstrate that the Bayesian complexity can be
used to assess how many effective parameters a set of data can support and that
it is a useful complement to the model likelihood (the evidence) in model
selection questions. We apply this approach to recent measurements of cosmic
microwave background anisotropies combined with the Hubble Space Telescope
measurement of the Hubble parameter. Using mildly non-informative priors, we
show how the 3-year WMAP data improves on the first-year data by being able to
measure both the spectral index and the reionization epoch at the same time. We
also find that a non-zero curvature is strongly disfavored. We conclude that
although current data could constrain at least seven effective parameters, only
six of them are required in a scheme based on the Lambda-CDM concordance
cosmology.
