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Inferences from optimal filtering equation. (English) Zbl 1327.60097
Summary: The processing of stationary random sequences under nonparametric uncertainty is given by a filtering problem when the signal distribution is unknown. A useful signal $$(S_n)_{n\geq1}$$ is assumed to be Markovian. This assumption allows us to estimate the unknown $$(S_n)$$ using only an observable random sequence $$(X_n)_{n\geq1}$$. The equation of optimal filtering of such a signal has been obtained by A. V. Dobrovidov [Autom. Remote Control 44, No. 6, 757–768 (1983); translation from Avtom. Telemekh. 1983, No. 6, 85–98 (1983; Zbl 0539.93076)]. Our result states that when the unobservable Markov sequence is defined by a linear equation with Gaussian noise, the equation of optimal filtering coincides with both the classical Kalman filter and the conditional expectation defined by the theorem on normal correlation.

##### MSC:
 60G35 Signal detection and filtering (aspects of stochastic processes) 60J05 Discrete-time Markov processes on general state spaces 62M20 Inference from stochastic processes and prediction 62M05 Markov processes: estimation; hidden Markov models
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