Quantifying galactic clustering and departures from randomness of the inter-galactic void probability function using information

A number of recent studies have estimated the inter-galactic void probability function and investigated its departure from various random models. We study a family of parametric statistical models based on gamma distributions, which do give realistic descriptions for other stochastic porous media. Gamma distributions contain as a special case the exponential distributions, which correspond to the ‘random’ void size probability arising from Poisson processes. The random case corresponds to the information-theoretic maximum entropy or maximum uncertainty model. Lower entropy models correspond on the one hand to more ‘clustered’ structures or ‘more dispersed’ structures than expected at random. The space of parameters is a surface with a natural Riemannian structure, the Fisher information metric. This surface contains the Poisson processes as an isometric embedding and provides the geometric setting for quantifying departures from randomness and perhaps on which may be written evolutionary dynamics for the void size distribution. Estimates are obtained for the two parameters of the void diameter distribution for an illustrative example of data published by Fairall.