A new phenomenological Helmholtz free energy function is proposed and afterwards revisited within the context of molecular-statistical theory to reconcile both theories of developing hyperelastic constitutive models. The functional is based on an inverse polynomial, which was developed by Nelder to describe processes in agricultural systems. In order to demonstrate the high adaptability of the functional describing structural silicones, rubber-like materials as well as polymeric glass laminate interlayers, six different materials are approximated under different deformation states. The parameter space of the material parameters to be calibrated is limited in order to ensure stable material behaviour under any deformations by verifying polyconvexity. The material parameter identification (and thus model calibration) is embedded within a Bayesian inverse problem formulation to obtain the posterior distributions of the material parameters involved. The Bayesian setup naturally considers the polyconvexity conditions by specifying respective prior distributions for the parameters. Besides that a probabilistic model selection is possible through the comparison of the evidence of the Extended Tube model against the evidence of the new energy function for the obtained data sets. The Bayesian model calibration approach further allows to estimate the predictive uncertainty for the material models under investigation for future deformation cases.