Perhaps the most extensive pursuit of this theme is to be found in Gutowitz, Victor, and Knight's local structure theory. It differs from the approach of Wilbur et.al. by their use of the theory of probability measures to justify the derivation of their equations for the probabilities of the blocks, even though the final equations are the same. Both groups of authors survey several classes of rules, comparing the theoretical results with empirically observed frequencies. There is general agreement that the probability of a block in one generation should equal the probability of its ancestor in the previous generation; differences arise both from the philosophies and the actual techniques used to obtain the required probability.
To derive their equations, Gutowitz et.al. require 2r+1 consecutive overlapping n-blocks to build up the n+2r-block ancestor of a given n-block from other n-blocks. The choice of a notation in which to express the equations is important; we shall explore certain alternatives. To begin with, suppose that B and X are words, the remaining symbols single letters. The equations for self consistent probabilities then read
As Gutowitz et.al. demonstrated, these equations reduce to the mean field equations for 1-block probabilities, making their local field theory a plausible extension of mean field theory. Since the 1-block, or mean field theory, denominators reduce to the constant value the mean field theory equations are polynomial equations, whereas the general local structure theory equations involve rational fractions. As a result it is much harder to establish the existence, uniqueness, or stability of their solutions.
Nevertheless, the local structure theory approach makes contact with a very general framework within probability theory, and also presents the basic equations in a symmetrical form admitting a greater variety of interpretations. In particular we can recover the point of view of Wilbur et.al. , interpreting the equations as a set of linear equations whose matrix of coefficients depends on the same solutions which it defines. The nonlinearity of the equations is thus of a very special form, reminiscent of the Hartree equations or the Hartee-Fock equations of quantum mechanics, which are generally solved by an iterative process.
Numerically, the equations of Gutowitz et.al. can be solved exclusively by iteration. Diagonalizing the coefficient matrix might accelerate convergence, but at the cost of the time expended on diagonalization. It is of greater importance that the form of the equations reveals particular properties of the solution and its convergence that are not otherwise evident.
The ostensible variables are not manifest in the denominators of the equations, but they are readily obtained by using the Kolmogorov consistency conditions. Since the terms of the numerators are associated with links in the de Bruijn diagram, and the denominators with nodes, the whimsically minded might fancy that they see a resemblance to the propagators and Feynman diagrams of field theory. In any event we are going to take up a slightly different interpretation.