What do we learn from computer simulations of Bell experiments?

Contrary to counterfactual definiteness quantum theory teaches us that measuring instruments are not passively reading predetermined values of physical observables. Counterfactual definiteness allows proving Bell inequalities. If the contextual character of quantum measurements is correctly taken into account the proofs of these inequalities may not be done. In recent computer simulations of idealized Bell experiment predetermined successive outcomes of measurements for each setting and predetermined time delays of their registrations are calculated. Time windows and time delays are used to select various samples. Correlations, estimated using these selected samples are consistent with the predictions of quantum theory and the time window dependence is similar to the dependence observed in some real experiments. It is an important example how correlations can be explained without evoking quantum non-locality. However by using a suitable post-selection one may prove anything. Since before the post-selection generated samples may not violate Bell inequalities as significantly as finite samples generated using quantum predictions thus one may not conclude that counterfactual definiteness is not able to distinguish classical from quantum physics. Moreover we show that for each choice of a time window there exists a contextual hidden variable probabilistic model consistent with the post-selection procedure used by the authors what explains why they are able to reproduce quantum predictions.