Quantifying dimensionality: Bayesian cosmological model complexities

We demonstrate a measure for the effective number of parameters constrained by a posterior distribution in the context of cosmology. In the same way that the mean of the Shannon information (i.e., the Kullback-Leibler divergence) provides a measure of the strength of constraint between prior and posterior, we show that the variance of the Shannon information gives a measure of dimensionality of constraint. We examine this quantity in a cosmological context, applying it to likelihoods derived from the cosmic microwave background, large-scale structure and supernovae data. We show that this measure of Bayesian model dimensionality compares favorably both analytically and numerically in a cosmological context with the existing measure of model complexity used in the literature.