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Filtering the Wright-Fisher diffusion. (English) Zbl 1181.93084

Summary: We consider a Wright-Fisher diffusion \((x(t))\) whose current state cannot be observed directly. Instead, at times \(t_{1} < t_{2} < \dots \), the observations \(y(t_i)\) are such that, given the process \((x(t))\), the random variables \((y(t_i))\) are independent and the conditional distribution of \(y(t_i)\) only depends on \(x(t_i)\). When this conditional distribution has a specific form, we prove that the model \(((x(t_i),y(t_i)), i\geq 1)\) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations.

MSC:

93E11 Filtering in stochastic control theory
60G35 Signal detection and filtering (aspects of stochastic processes)
62C10 Bayesian problems; characterization of Bayes procedures
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