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Approximating marginals of a graphical model is one of the fundamental problems in the theory of networks. In a recent paper a method was shown to construct a variational free energy such that the linear response estimates, and maximum entropy estimates (for beliefs) are in agreement, with implications for direct and inverse Ising problems[1]. In this paper we demonstrate an extension of that method, incorporating new information from the response matrix, and we recover the adaptive-TAP equations as the first order approximation[2]. The method is flexible with respect to applications of the cluster variational method, special cases of this method include Naive Mean Field (NMF) and Bethe. We demonstrate that the new framework improves estimation of marginals by orders of magnitude over standard implementations in the weak coupling limit. Beyond the weakly coupled regime we show there is an improvement in a model where the NMF and Bethe approximations are known to be poor for reasons of frustration and short loops.
We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to t
The discrete polymer model with random Boltzmann weights with homogeneous inverse gamma distribution, introduced by Seppalainen, is studied in the case of a polymer with one fixed and one free end. The model with two fixed ends has been integrated by
We present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities, for a large class of mean field disordered models, as for example the Sherrington-Kirkpatrick model, and the Derrida
We consider the complexity of random ferromagnetic landscapes on the hypercube ${pm 1}^N$ given by Ising models on the complete graph with i.i.d. non-negative edge-weights. This includes, in particular, the case of Bernoulli disorder corresponding to
The phase diagram of the random field Ising model on the Bethe lattice with a symmetric dichotomous random field is closely investigated with respect to the transition between the ferromagnetic and paramagnetic regime. Refining arguments of Bleher, R