Paper Title
Explicitly Sample-Equivalent Dynamic Models for Gaussian Markov, Reciprocal, and Conditionally Markov Sequences

Abstract
The conditionally Markov (CM) sequence contains different classes including Markov, reciprocal, and so-called CML and CMF (two special classes of CM sequences). Each class has its own forward and backward dynamic models. The evolution of a CM sequence can be described by different models. For example, a Markov sequence can be described by a Markov model, as well as by reciprocal, CML, and CMF models. Also, sometimes a forward model is available, but it is desirable to have a backward model for the same sequence (e.g., in smoothing). Therefore, it is important to study relationships between different dynamic models of a CM sequence. This paper discusses such relationships between models of nonsingular Gaussian (NG) CML, CMF , reciprocal, and Markov sequences. Two models are said to be explicitly sample-equivalent if not only they govern the same sequence, but also a one-one correspondence between their sample paths is made explicitly. A unified approach is presented, such that given a forward/backward CML/CMF /reciprocal/Markov model, any explicitly equivalent model can be obtained. As a special case, a backward Markov model explicitly equivalent to a given forward Markov model can be obtained regardless of the singularity/non singularity of the state transition matrix of the model. Keywords - Conditionally Markov, reciprocal, Markov, Gaussian sequence, dynamic model, explicitly sampleequivalent.