Gaussian conditionally markov sequences: Algebraically equivalent dynamic models

The conditionally Markov (CM) sequence contains different classes, including Markov, reciprocal, and so-called CM_L and CM_F (two CM classes defined in our previous work). Markov sequences are special reciprocal sequences, and reciprocal sequences are special CM_L and CM_F sequences. Each class has its own forward and backward dynamic models. The evolution of a CM sequence can be described by different models. For a given problem, a model in a specific form is desired or needed, or one model can be easier to apply and better than another. Therefore, it is important to study the relationship between different models and to obtain one model from another. This article studies this topic for models of nonsingular Gaussian CM_L, CM_F, reciprocal, and Markov sequences. Two models are probabilistically equivalent (PE) if their stochastic sequences have the same distribution and are algebraically equivalent (AE) if their stochastic sequences are pathwise identical. A unified approach is presented to obtain an AE forward/backward CM_L/CM_F/reciprocal/Markov model from another such model. As a special case, a backward Markov model AE to a forward Markov model is obtained. While existing results are restricted to models with nonsingular state transition matrices, our approach is not. In addition, a simple approach is presented for studying and determining Markov models, whose sequences share the same reciprocal/CM_L model.