This research work concerns the stochastic modeling and statistical inverse identification of a phase-field model for fracture in random heterogeneous elastic materials exhibiting almost surely (a.s.) isotropic symmetry properties. Within the framework of linear elasticity theory and probability theory, a stochastic model for a.s. isotropic random elasticity fields adapted to standard phase-field models for brittle fracture is proposed and constructed using the maximum Entropy (MaxEnt) principle. In the present work, we consider a standard phase-field model for brittle fracture of isotropic elastic materials which is classically parameterized by the fracture toughness (or critical energy release rate) and the regularization parameter (corresponding to the actual width of phase-field/damage localization band). A sensitivity analysis is carried out to study the influence of some fracture properties (fracture toughness) and some hyperparameters (dispersion coefficient and spatial correlation length) involved in the stochastic model of random elasticity field on the crack path and the global force-displacement response. Finally, a statistical inverse identification method based on a nonparametric Bayesian approach is used to estimate the posterior probability distribution of the random fracture toughness. The proposed approach is illustrated on a classical benchmark problem for brittle fracture of a two-dimensional single-edge notched/cracked square specimen undergoing either uniaxial tension (mode I) or pure shear (mode II) loading.