Probability Models and The Kolmogorov Structure Function
For every computable probability distribution P it can be proved that . For example, if P is the uniform distribution on the set S of strings of length n, then each has probability . In the general case of computable probability mass functions we incur a logarithmic additive error term. Kolmogorov's structure function becomes
where x is a binary string of length n with where P is a contemplated model (computable probability of n-length strings) for x, is the Kolmogorov complexity of P and is an integer value bounding the complexity of the contemplated P's. Clearly, this function is nonincreasing and reaches for where c is the required number of bits to change x into and is the Kolmogorov complexity of x. Then . For every complexity level the function is the Kolmogorov complexity version of the maximum likelihood (ML).
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