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Complexity of models
The pictures below show least squares fits (found using Fit in Mathematica) to polynomials with progressively higher degrees and therefore progressively more parameters. Which fit should be considered best in any particular case must ultimately depend on external considerations. But since the 1980s there have been attempts to find general criteria, typically based on maximizing quantities such as -Log[p] - d (the Akaike information criterion), where p is the probability that the observed data would be generated from a given model ( -Log[p] is proportional to variance in a least squares fit), and d is the number of parameters in the model.

The pattern produced continues to expand on both left and right, but only the part that fits across the page is shown here.

The pattern continues to expand on the left forever, but only the part that fits across each page is shown.

The pattern continues to expand on the left forever, but only the part that fits across each page is shown.

The pattern continues to expand on the left forever, but only the part that fits across each page is shown.

In the pictures shown here, every replacement that is found to fit in a left-to-right scan is performed at each step.

For example, what I say about the fundamental similarity of human thinking to other processes in nature may seem to fit with Buddhism. And what I say about the irreducibility of processes in nature to short formal rules may seem to fit with Taoism.

Self-assembly
Given elements (such as pieces of molecules) that fit together only when certain specified constraints are satisfied it is fairly straightforward to force, say, cellular automaton patterns to be generated, as on page 221 .