the diagonal called sufficiency line L defined by . It is approached to within a constant distance by the graph of for certain arguments (for instance, for ). For these 's we have and the associated model (witness for ) is called an optimal set for , and its description of bits is therefore an algorithmic sufficient statistic. We write `algorithmic' for `Kolmogorov complexity' by convention. The main properties of an algorithmic sufficient statistic are the following: If is an algorithmic sufficient statistic for , then . That is, the two-part description of using the model and as data-to-model code the index of in the enumeration of in bits, is as concise as the shortest one-part code of in bits. This can be easily seen as follows: , using straightforward inequalities and the sufficiency property, we find that . (For example, given , we can describe self-delimitingly (you can determine its end) in bits.) Therefore, the randomness deficiency of in is a constant, which means that is a typical (random) element of S. However, there can be models containing that are not sufficient statistics. An algorithmic sufficient statistic for has the additional property, apart from being a model of best fit, that and therefore by the Kolmogorov complexity symmetry of information (the information about in is about the same as the information about in x) we have : the algorithmic sufficient statistic is a model of best fit that is almost completely determined by . ( is a shortest program for .) The algorithmic sufficient statistic associated with the least such is called the algorithmic minimal sufficient statistic. With respect to the picture: The MDL structure function is explained below. The Goodness-of-fit structure function is the least randomness deficiency (see above) of any model for such that . This structure function gives the goodness-of-fit of a model (containing x) for the string x. When it is low the model fits well, and when it is high the model doesn't fit well. If for some then there is a typical model for such that and is typical (random) for S. That is, is the best-fitting model for x. For more details see and especially and. Selection of properties Within the constraints that the graph goes down at an angle of at least 45 degrees, that it starts at n and ends approximately at , every graph (up to a additive term