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In this paper we focus on the semantics of undirected graphical model. We present a general specification based on exponential family distributions that allows great model flexibility and leads to consistent inferential procedures. The model is extended to include prior distributions on the parameters, that reduce the variance of the estimates and permit to avoid over parametrization. Particular attention is devoted to non-differentiable $l_{1}$ penalization, that leads to non-explicit gradient, for which we propose a new differentiable approximation. Experimental results and applications to large scale data are provided to demonstrate the increase in the rate of convergence and the variance reduction for different type of penalization priors.