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Optimal filtering and the dual process

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Advanced Monte Carlo Methods for Complex Inference Problems

Co-author: Matteo Ruggiero (Turin)

We link optimal filtering for hiddenMarkov models to the notion of duality forMarkov processes.We show that when the signal is dual to a process that has two components, one deterministic and one a pure death process, and with respect to functions that define changes of measure conjugate to the emission density, the filtering distributions evolve in the family of finite mixtures of such measures and the filter can be computed at a cost that is polynomial in the number of observations. Special cases of our framework include the Kalman filter, and computable filters for the CoxIngersollRoss process and the one-dimensional Wright Fisher process, which have been investigated before in the literature. The dual we obtain for the Cox IngersollRoss process appears to be new in the literature.

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