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-pk, Eph+k, E-dh, E-dk, Edh+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 thre e-phase structure invariants are derived, showing how the reliability of th e probabilistic estimates depends on the errors.