Dynamic Inference

We consider a model for dynamic uncertain processes that affords considerably more generality of formulation than do Markovian models or their derivatives. The underlying statistical parameters of a stochastic process that produces observable outputs are themselves allowed to change at times generated by another stochastic process. We would like to make probability assignments to future outputs of the process, given only the past outputs. We develop the inferential relations for the case where the changes of parameters are governed by a renewal process, and where the process that generates observables depends only on its present parameters. We illustrate these results using an example with a Bernoulli observable distribution, a beta parameter distribution, and a geometric distribution for for the time between parameter changes. The numerical results indicate a complexity of behavior that challenges intuition. Possible applications of the general class of dynamic inference models range from marketing to anti-submarine warfare.