This research work concerns the forward numerical simulation and statistical inverse identification of a phase field model for fracture in random heterogeneous elastic materials. Within the framework of linear elasticity theory and probability theory, a stochastic model for almost surely (a.s.) isotropic random elastic materials adapted to standard phase field models for brittle fracture is proposed and constructed using the maximum Entropy (MaxEnt) principle. A sensitivity analysis is carried out to study the influence of some fracture properties (fracture toughness) and random spatially-varying elastic material properties on the crack path and the global force-displacement response. Finally, a data-driven statistical inverse identification method based on a nonparametric Bayesian approach is proposed to estimate the posterior probability distribution of the random fracture toughness. The proposed approach is illustrated on two classical benchmark problems for brittle fracture, namely a mode I fracture problem (uniaxial tension test) and a mode II fracture problem (pure shear test) of a two-dimensional single-edge notched/cracked square specimen.