Equivalence classes of functions of finite Markov chains

A matrical representation of a Markov chain consists of the initial vector and transition matrix of the chain, along with matrices that specify which observable response occurs for each state. The likelihood function based on a Markov model can be stated in a general way using the components of the model's matrical representation. It follows directly from that statement that two models are equivalent in likelihood if they are related through matrix operations that constitute a change of basis of the matrical representation. Two necessary properties of a change matrix associating two Markov models that are members of the same equivalence class with respect to likelihood are derived. Examples are provided, involving use of the results in analyzing identifiability of Markov models, including a useful application of diagonalization that provides a connection between the problem of identifiability and the eigenvalue problem.