  In this work we introduce a new and richer class of finite order Markov chain
models and address the following model selection problem: find the Markov model
with the minimal set of parameters (minimal Markov model) which is necessary to
represent a source as a Markov chain of finite order. Let us call $M$ the order
of the chain and $A$ the finite alphabet, to determine the minimal Markov
model, we define an equivalence relation on the state space $A^{M}$, such that
all the sequences of size $M$ with the same transition probabilities are put in
the same category. In this way we have one set of $(|A|-1)$ transition
probabilities for each category, obtaining a model with a minimal number of
parameters. We show that the model can be selected consistently using the
Bayesian information criterion.
