The Joint Probability Distribution of Structure Factors

This section is based Bricogne (1984) section 4.

Our aim is to determine the joint probability distribution of structure factors, i.e. the probability of the set of all structure factors taking on a particular set of values, given some prior assumption about the distribution of atoms. From this we can generate a conditional probability distribution of some subset of structure factors given some known or assumed values of the remainder. Alternatively, we can generate a marginal probability distribution of some subset by integrating out structure factors which are unknown.

Let this joint distribution be called , where is a vector of structure factors:

Note that given a knowledge of all the atomic positions we can assemble in two ways:

1. We can calculate the contribution from each atom to the total scattering, and then accumulate all the contributions (as is done in the structure factor equation).
2. We can build a map by accumulating the contributions from each atom to the electron density, and then take the Fourier transform of this map.

Similarly, we may consider the joint probability distribution of structure factors as a joint probability distribution of density values in a map. The probability of a particular set of structure factors (given some prior distribution of atoms) is equal to the probability of the corresponding density distribution (given the unitarity of the Fourier Transform).

For simplicity we will consider a structure containing a single atomic type with no thermal motion, thus the structure may be represented by a single probability distribution function of atomic coordinates whose Fourier Transform is a set of unitary structure factors .

Next, we divide up the unit cell into B boxes. The probability of an atom being placed in box j is (where q is our prior estimate of the probability distribution of atoms). A configuration of the map can be described by the number of atoms in each box. Then the probability of a configuration is given by the number of ways a configuration can be achieved multiplied by the probability of generation of a certain number of atoms in each box:

where . Expanding the factorials in terms of Stirling's formula gives:

where . The summation in this expression is simply the entropy of the probability density. In the continuous form the expressions are as follows:

How should we construct the prior probability ? This should be the most likely probability density function consistent with whatever data is available, in other words the probability density function with the highest probability, and therefore entropy, consistent with any known structure factors and/or density constraints.

Thus the procedure for determining the joint probability of a set of structure factors is as follows:

1. Determine the maximum entropy probability density function consistent with the given constraints.
2. Calculate the logarithm of the relative entropy of this probability density.