Quantifying dimensionality: Bayesian cosmological model complexities

TL;DR

The paper tackles the challenge of quantifying how many parameters are effectively constrained by cosmological data in the presence of many nuisance parameters. It proposes Bayesian model dimensionality (BMD), defined as the variance of the Shannon information under the posterior, as a robust, estimator-free alternative to the traditional Bayesian model complexity. Through analytical tests and real cosmological data (Planck, DES, SH0ES, BAO), it shows that BMD yields intuitive and stable dimensionalities, while conventional BMC estimates can be unstable and parameterisation-dependent. Thermodynamic interpretation and practical applications (shared dimensionalities, information criteria) further connect BMD to broader statistical concepts. The work advocates routinely reporting the triple (evidence, KL divergence, BMD) and demonstrates that BMD provides a practical, principled tool for understanding tension and information content in cosmological analyses.

Abstract

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 Cosmic Microwave Background, large scale structure and supernovae data. We show that this measure of Bayesian model dimensionality compares favourably both analytically and numerically in a cosmological context with the existing measure of model complexity used in the literature.

Figures (9)

• Figure 1: Distributions with the same Kullback-Leibler divergence, but differing dimensionalities. Both the right and left-hand plots indicate two-dimensional probability distributions. In each plot, the lower left panel is a two-dimensional contour plot indicating the iso-probability contours enclosing $66\%$ and $95\%$ of the probability mass. The upper and lower right panels indicate the one-dimensional marginal probability distributions. There is an implicit uniform prior over the ranges indicated by the axis ticks.
• Figure 2: The typical set of a $d$-dimensional Gaussian distribution can be visualised by plotting the posterior probability distribution of the Shannon information $\mathcal{I}$. The posterior has mean $\mathcal{D}$ and variance $\frac{d}{2}$. The posterior maximum occurs at $\mathcal{I}=\mathcal{D}+1$, and the domain is $(-\infty,\mathcal{I}_{\max}]$. The above plot is shown for $d=16$ in analogy with the Planck likelihood from \ref{['fig:shannon_examples']} and \ref{['tab:numerics']}.
• Figure 3: Bayesian dimensionality for the common one-dimensional distributions in \ref{['tab:analytics']}. Widths are normalised so that the distributions all have the same Kullback-Leibler divergence $\mathcal{D}$. The dashed curve in all plots is a Gaussian distribution.
• Figure 4: Dependency of dimensionality and Kullback-Leibler divergence on prior volume for a Cauchy distribution $\mathcal{P}(x) \propto (1+x^2)^{-1}$. Whilst the BMD and BMC are pathologically large $(\gg 1)$ if the full domain of the Cauchy distribution is included, truncating the range to a lower prior volume $x\in [-V/2,V/2]$ reduces the dimensionality to more sensible values.
• Figure 5: Cosmological parameters unconstrained by DES. Whilst DES provides constraints on four of the cosmological parameters, it tells us nothing of $\tau$, and little of a correlated combination of $\ln 10^{10} A_s$ and $n_s$. This figure should be compared with \ref{['fig:dimensions']}.