The Kalman filter (KF) methodology is apt to solve parameter identification (inverse) problems in a statistical context, through a sequence of estimations, which starts from an a priori estimation by an ''expert'' (Bayesian approach) and exploits a time-stepping flow of experimental data until convergence is empirically ascertained. Such methodology is here adopted for the identification of the material parameters, together with their uncertainties, in a mode I cohesive crack model, on the basis of experimental data generated by wedge-splitting tests on concrete specimens. The simulation of the experiments is based on the assumptions of a piecewise-linear cohesive model with four parameters to identify on the crack path, and of linear elasticity elsewhere. In view of regularly progressive fracture processes, the discrete crack model and, consequently, the overall finite element analysis, are formulated as linear complementarity problems. This mathematical construct is exploited to obtain the sensitivity matrix, key ingredient of the KF extended to nonlinear inverse problems, in a computationally convenient closed form. Various issues peculiar of KF identification in the present mechanical context are critically discussed in the light of the numerical solutions achieved.