Structure Function

Abstract

We extend Kolmogorov’s structure function in the algorithmic theory of complexity to statistical models in order to avoid the problem of noncomputability. For this we have to construct the analog of Kolmogorov complexity and to generalize Kolmogorov’s model as a finite set to a statistical model. The Kolmogorov complexity K(xⁿ) will be replaced by the stochastic complexity for the model class $\mathcal{M}_\gamma$ _γ (5.40) and the other analogs required will be discussed next. In this section the structure index γ will be held constant, and to simplify the notations we drop it from the models written now as f(xⁿ;ϑ), and the class as $\mathcal{M}_k$ _k. The parameters θ range over a bounded subset ω of R^k.