large number of filtering procedures, researchers encounter the necessity to have a quantitative methodology able to estimate the information retained in a filtered correlation matrix obtained from the sample correlation matrix. It is also important to quantify the stability of the filtering procedure in different realizations or replicas of the process and a distance of the filtered correlation matrix from a given reference model. For all the above listed purposes, a very useful measure is the one using the Kullback-Leibler distance that was introduced in Tumminello et al. (2007a). This distance presents the important characteristics that its value quantifying the distance between a sample correlation matrix and the correlation matrix of the generating model turns out to be independent from the specific correlation matrix of the model both for multivariate Gaussian variables (Tumminello et al., 2007a) and for multivariate Student’s $t$ variables (Biroli et al., 2007; Tumminello et al., 2007b).

In the present paper we discuss in a coherent and self-consistent way (i) some filtering procedures of the correlation matrix based on hierarchical clustering and the bootstrap validation of hierarchical trees and correlation based networks, (ii) the hierarchically nested factor model, (iii) the Kullback-Leibler distance between the probability density functions 