Some comments on computational mechanics, complexity measures, and all that

We comment on some conceptual and and technical problems related to computational mechanics, point out some errors in several papers, and straighten out some wrong priority claims. We present explicitly the correct algorithm for constructing a minimal unifilar hidden Markov model ("$\epsilon$-machine") from a list of forbidden words and (exact) word probabilities in a stationary stochastic process, and we comment on inference when these probabilities are only approximately known. In particular we propose minimization of forecasting complexity as an alternative basis for statistical inference of time series, in contrast to the traditional maximum entropy principle. We present a simple and precise way of estimating excess entropy (aka "effective measure complexity". Most importantly, however, we clarify some basic conceptual problems. In particular, we show that there exist simple models (called "totally recurrent graphs") where none of the nodes of the "$\epsilon$-machine" (the "causal states") corresponds to an element of a state (or history) space partition.