Electrical Impedance Tomography (EIT) of closed conductive media is an ill-posed inverse problem. In order to solve the corresponding direct problem, the Finite Elements Method (FEM) provides good accuracy and preserves the non linear dependence of the observation set upon the conductivity distribution. In this paper, we show that the Bayesian approach presented in [1] for linear inverse imaging problems is also valid for a non linear problem such as EIT. Our contribution is based on an edge-preserving Markov model as prior for conductivity distribution. Maximum a posteriori reconstruction results from 40 dB noisy measurements (simulated with a finer mesh) yield significant resolution improvement compared to classical methods.