Measuring the effective complexity of cosmological models

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 noninformative 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 nonzero 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 $\Lambda \mathrm{CDM}$ concordance cosmology.

Figure 1

Bayesian effective complexity $C_b$ (solid black line, left-hand vertical scale) and model likelihood (red circles, right-hand scale) as a function of the number of parameters, for $d=10$ data points with small noise. The dashed blue line is the number of parameters for reference. The errorbars on the model likelihood values are smaller than the symbols on this scale, while the Bayesian complexity is independent of the noise realization (i.e., error-free) for linear models. The Bayesian analysis correctly concludes that the best model is the one with $n=6$ parameters.

Figure 2

As in Fig. 1, but now using only $p=4$ data points. The maximum effective complexity that the data can support is ${\mathcal{C}}_b=4$, and the flattening of the model likelihood at the same value does not allow to conclude that models with more parameters are disfavored.

Figure 3

As in Fig. 1, but now with $p=10$ data points and large noise. As in Fig. 2, the maximum complexity supported by the data is smaller than the underlying true model complexity, $m=6$.

Figure 4

We plot the model likelihood (normalized to the model with the most parameters) versus the Bayesian effective complexity for the models of Table  (using WMAP 3 yr data). A downward-pointing arrow indicates the Bayesian complexity of models that lie outside the boundary of the figure.