On the Concept of Importance in Markoffian Processes

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Abstract
The concept of “forward importance” in neutron physics is extended to more general evolution systems subject to Markoffian processes. The evolution systems are assumed to interact with external environments such as observation systems. The forward importance represents an expected value of the amount of the interactions of a stochastic system with external environment in the process of its temporal evolution. Especially, if an evolution system is interacting with observation system, the forward importance represents an expected amount of the contributions of the evolution system to the meter-reading of the observation system. The forward importance enables us to choose the most suitable initial state for finally conditioned purpose in stochastic sense. A new type of importance “backward importance” is introduced into the time-reversed Markoffian process, which represents a retrodicted value of the amount of the interactions of an evolution system with external environment. Here the retrodiction is made with the use of the posteriori (Bayes') probability. The forward (or backward) importance and the forward (or backward) transition probability density function are shown to be treated as canonically conjugate variables in a variational formulation of the Markoffian process. It is shown that the H- (or inverse H-) theorem can be viewed as the conservation law for the forward (or backward) importance.

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