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Kolmogorov consistency conditions

The motivation for working with block probabilities rather than cell probabilities is the hope that a better agreement with empirical observations can be obtained, supposing that up to some point longer blocks can better account for correlations between cells than short blocks can. Nevertheless, the empirical quantity which is usually calculated is the probability of individual cells. Sometimes variances or pair probabilities may also be calculated, but it is the density of cells which is the primary concern.

Figure: Two-block probabilities summed from four-block probabilities.  

Cell density can be inferred from block probabilities by summing up the probabilities of blocks in various ways, all of which are guaranteed to give consistent results by virtue of the Kolmogorov consistency conditions. It is thus worthwhile to investigate whether the iterative solution of the local field theory equations conserves the consistency conditions. We have shown that if the definition of -block probabilities in terms of n-block probabilities is consistent, the definitions for all shorter blocks will also be consistent. Thus it suffices to show that one cycle of iteration conserves consistency at the the highest level---the one which is anyway involved in defining the denominators for .

Wilbur included a proof in their article, so the only practical question remaining is one of stability---whether numerical errors arising during the course of an iterative solution of the equations could prejudice the consistency of the results.

It is in any event instructive to write the field equations in a very extended explicit matrix form, shown here for 2-blocks evolving by Rule 22. We begin with equations (set aside in Figure gif due to their bulk) expressing the probability of a block as the sum of the probabilities of its possible ancestors.

Either Wilbur 's theory or Gutowitz 's equations written in Hartree-Fock form estimate the probability of each 4-block ancestor in terms of the probabilities of the 1- and 2-block segments into which the ancestor can be decomposed. The matrix form of this estimate, in which the precursor of the de Bruijn format is quite apparent, is shown in Figure gif.

Figure: Four-block probabilities estimated from two-block probabilities.  

Consolidating these two equations produces a matrix of coefficients

in which it can be verified by inspection that is consistently defined by either or by or that equivalently on the right hand side implies the same relation for the new values on the left hand side of the equation.

It is easier to give the general proof in symbolic form, having first written the field equations in terms of the merged product:

If it is intended to write Q=AB so as to obtain

we need to know that

If is incorporated into X we introduce W

and note that

so that

Since an entirely symmetrical expression results when Q is factored into Q=BC, comparing the two establishes the consistency of after iteration, supposing that it was consistent before. The essential points in the proof are, first, that running through all initial letters in the sum for guarantees a traversal of all the initial letters in the ancestors, and second, that the denominators in the probabilities are cancelled by the sums in the numerators as a consequence of the consistency hypothesis.

next up previous contents
Next: The vector subset Up: Probabilistic evolution matrix Previous: Hartree-Fock approach

Harold V. McIntosh