Viscoelastic materials are widely utilized in engineering for their advantageous damping properties, which enable effective energy dissipation. They are also valued for their stiffness properties, particularly in the case of suspension design. The behavior of these materials is complex, making it challenging to accurately incorporate them into simulations. If the dependence of dynamic properties on frequency and temperature has been commonly investigated, other phenomena still need to be studied, as for instance nonlinear phenomena leading to changes with the magnitude of stress/strain. Amongst them, the Payne effect, which manifests itself by a monotonous decrease of the storage modulus, remains difficult to integrate in the simulations, and is not often characterized experimentally for vibration purposes. An approach is proposed to identify the Payne effect by combining tests derived from the Oberst beam set-up and numerical simulations. To take into account the uncertainties inherent to the experiments, the identification process is based on a Bayesian identification framework which allows to determine the parameters and their statistical properties from a limited number of experimental measurements. The coupling between the Bayesian identification of dynamic properties of a polymer sample and a digital twin of the experimental set-up allows to identify the evolution of the storage modulus as a function of the strain amplitude, as well as the confidence interval around the mean value. The application on a silicone sample confirmed the decrease of the modulus with the strain level, with a good confidence level. The methodology developed here for the specific case of the Payne effect can be extended more generally to the experimental characterization of nonlinear behavior.