Modern statistical modeling is an important complement to the more traditional approach of physics where Complex Systems are studied by means of extremely simple idealized models. The Minimum Description Length (MDL) is a principled approach to statistical modeling combining Occams razor with Information Theory for the selection of models providing the most concise descriptions. In this work, we introduce the Boltzmannian MDL (BMDL), a formalization of the principle of MDL with a parametric complexity conveniently formulated as the free-energy of an artificial thermodynamic system. In this way, we leverage on the rich theoretical and technical background of statistical mechanics, to show the crucial importance that phase transitions and other thermodynamic concepts have on the problem of statistical modeling from an information theoretic point of view. For example, we provide information theoretic justifications of why a high-temperature series expansion can be used to compute systematic approximations of the BMDL when the formalism is used to model data, and why statistically significant model selections can be identified with ordered phases when the BMDL is used to model models. To test the introduced formalism, we compute approximations of BMDL for the problem of community detection in complex networks, where we obtain a principled MDL derivation of the Girvan-Newman (GN) modularity and the Zhang-Moore (ZM) community detection method. Here, by means of analytical estimations and numerical experiments on synthetic and empirical networks, we find that BMDL-based correction terms of the GN modularity improve the quality of the detected communities and we also find an information theoretic justification of why the ZM criterion for estimation of the number of network communities is better than alternative approaches such as the bare minimization of a free energy.
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