An alternative method for the calculation of joint probability distributions. Application to the expectation of the triplet invariant

This paper presents a completely new method for the calculation of expectations (and thus joint probability distributions) of structure factors or phase invariants. As an example, a first approximation of the expectation of the triplet invariant (up to a constant) is given and a complex number is obtained. Instead of considering the atomic vector positions or reciprocal vectors as the fundamental random variables, the method samples over all functions (distributions) with a given number of atoms and given Patterson function. The aim of this paper was to explore the feasibility of the method, so the easiest problem was chosen: the calculation of the expectation value of the triplet invariant in P1. Calculation of the joint probability distribution of the triplet is not performed here but will be done in the future.