On integrating direct methods and isomorphous-replacement techniques: triplet estimation and treatment of errors

The method of joint probability distribution functions has been generalized in order to include and treat different sources of error. The probability distributions of the isomorphous pairs (E_p, E_d) and of the two triples (E_ph, E_pk, E_ph+k, E_dh, E_dk, E_dh+k) are obtained, on the assumption that the lack of isomorphism and the errors in measurements cumulate on the E_d variables. The conditional distributions of the two-phase and the three-phase structure invariants are derived, showing how the reliability of the probabilistic estimates depends on the errors.