On the equivalence between minimal sufficient statistics, minimal typical models and initial segments of the Halting sequence

Abstract: It is shown that the length of the algorithmic minimal sufficient statistic of a binary string x, either in a representation of a finite set, computable semimeasure, or a computable function, has a length larger than the computational depth of x, and can solve the Halting problem for all programs with length shorter than the m-depth of x. It is also shown that there are strings for which the algorithmic minimal sufficient statistics can contain a substantial amount of information that is not Halting information. The weak sufficient statistic is introduced, and it is shown that a minimal weak sufficient statistic for x is equivalent to a minimal typical model of x, and to the Halting problem for all strings shorter than the BB-depth of x.