Bounds for Algorithmic Mutual Information and a Unifilar Order Estimator

Inspired by Hilberg's hypothesis, which states that mutual information between blocks for natural language grows like a power law, we seek for links between power-law growth rate of algorithmic mutual information and of some estimator of the unifilar order, i.e., the number of hidden states in the generating stationary ergodic source in its minimal unifilar hidden Markov representation. We consider an order estimator which returns the smallest order for which the maximum likelihood is larger than a weakly penalized universal probability. This order estimator is intractable and follows the ideas by Merhav, Gutman, and Ziv (1989) and by Ziv and Merhav (1992) but in its exact form seems overlooked despite attractive theoretical properties. In particular, we can prove both strong consistency of this order estimator and an upper bound of algorithmic mutual information in terms of it. Using both results, we show that all (also uncomputable) sources of a finite unifilar order exhibit sub-power-law growth of algorithmic mutual information and of the unifilar order estimator. In contrast, we also exhibit an example of unifilar processes of a countably infinite order and an algorithmically random oracle, for which the mentioned two quantities grow as a power law with the same exponent. We also relate our results to natural language research.