Complete Identification of Binary-Valued

The complete identification problem is to decide whether a stochastic process (Xt) 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 have been given all of which depend on fur- ther assumptions on the processes. Here we present a full, general solu- tion for binary-valued hidden Markov processes. Our approach is rooted in algebraic statistics hence 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 algebra. As a consequence, our solution provides immediate algorithmic access where tests come in form of elementary (linear) algebraic routines.