in argument and value) is realized by the structure function of some data x and vice versa. Where the graph hits the diagonal first the argument (complexity) is that of the minimum sufficient statistic. It is incomputable to determine this place. See. Main property It is proved that at each level of complexity the structure function allows us to select the best model for the individual string x within a strip of with certainty, not with great probability. The MDL variant The Minimum description length (MDL) function: The length of the minimal two-part code for x consisting of the model cost K(S) and the length of the index of x in S, in the model class of sets of given maximal Kolmogorov complexity , the complexity of S upper bounded by , is given by the MDL function or constrained MDL estimator: where is the total length of two-part code of x with help of model S. Main property It is proved that at each level of complexity the structure function allows us to select the best model S for the individual string x within a strip of with certainty, not with great probability. Application in statistics The mathematics developed above were taken as the foundation of MDL by its inventor Jorma Rissanen. Probability models For every computable probability distribution it can be proved that . For example, if is some computable distribution on the set of strings of length , then each has probability . Kolmogorov's structure function becomes where x is a binary string of length n with where is a contemplated model (computable probability of -length strings) for , is the Kolmogorov complexity of and is an integer value bounding the complexity of the contemplated 's. Clearly, this function is non-increasing and reaches for where c is the required number of bits to change into and is the Kolmogorov complexity of . Then . For every complexity level the function is the Kolmogorov complexity version of the maximum likelihood (ML). Main property It is proved that at each level of complexity the structure function allows us to select the best model for the individual string within a strip of with certainty, not with great probability. The MDL variant and probability models The MDL function: The length of the minimal two-part code for x consisting of the model cost K(P) and the length of , in the model