# Quantifying dimensionality: Bayesian cosmological model complexities

## TL;DR

This paper introduces a Bayesian measure of the effective number of parameters constrained by data, using the variance of Shannon information, and applies it to cosmological datasets to evaluate model complexity.

## Contribution

It proposes a new measure of Bayesian model dimensionality based on Shannon information variance and demonstrates its effectiveness in cosmological analyses.

## Key findings

- The measure aligns well with existing complexity metrics.
- It provides a robust way to quantify parameter constraints.
- Applied to CMB, large scale structure, and supernova data.

## 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.