Algebraic identification of binary-valued hidden Markov processes
Alexander Schoenhuth
The complete identification problem is to decide whether a stochastic process
$(X_t)$ is a hidden Markov process and if yes to infer a corresponding
parametrization. So far only partial answers to either the decision or the
inference part had been available all of which depend on further assumptions on
the processes which are usually centered around stationarity. Here we present a
full, general solution for binary-valued hidden Markov processes. Our approach
is rooted in algebraic statistics and therefore geometric in nature. We
demonstrate that the algebraic varieties which describe the probability
distributions associated with binary-valued hidden Markov processes are zero
sets of determinantal equations which draws a connection to well-studied
objects from (linear) algebra. Therefore our solution immediately gives rise to
algorithmic tests in form of elementary algebraic routines.