A procedure to identify the viscoelastic material parameters of a solid amorphous polymer and to estimate their values is presented. Stress-strain material data is obtained for the polymer by a compression experiment. The material behavior of the polymer is modeled according to the generalized Maxwell model, which is fitted to the experimental data by the method of least squares to obtain a first approximation for the model parameters. The identification of the model parameters is completed by a Markov chain Monte Carlo (MCMC) method, which generates the probability distributions of the relevant parameters of the material. The utilized MCMC method enables us to determine a suitable complexity (i.e., the number of Maxwell elements) for the generalized Maxwell model, so that the model best fits the data and, simultaneously, leads to an identifiable set of parameters. The numerical results imply that the uniqueness of the solution is lost when the number of model parameters becomes redundant.