We require the maximum entropy probability distribution consistent
with a set of structure factor constraints and some prior probability
, which may represent a molecular envelope or some other prior
information. Let the desired distribution be 
, whose
Fourier coefficients are 
.
If M structure factors are known both in magnitude and phase, these may be expressed as complex constraints of the following form
where 
, 
. We include the origin term with j=0, 
.
We must find the function 
consistent with the constraints
(6) for which the entropy 
is a maximum. Using the method
of Lagrange multipliers, this is equivalent to the unconstrained
maximisation of the function
with respect to 
, 
.
Differentiating with respect to the 
returns the constraint
equations (6), while differentiating with respect to
yields the equations:
or:
We may solve for 
directly by integrating this equation
over 
and using 
(constraint 0):
Then:
Equations (11), (12), and (13) are the
maximum entropy equations. These must be solved for the Lagrange
multipliers 
, and thus determine the maximum entropy
distribution 
.
