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Table 4 Adaptation based on weighted samples

From: A survey of Monte Carlo methods for parameter estimation

Let us compute the set of IS weights (potentially different from those of Eq. (71)),
\({\rho }_{n,t}^{(m)} = \frac {\pi (\boldsymbol {\theta }_{n,t}^{(m)})}{\Omega _{n,t}(\boldsymbol {\theta }_{n,t}^{(m)})},\)
where Ωn,t are chosen in such a way that they do not jeopardize the consistency of the IS estimators, and they can be equal to \(w_{n,t}^{(m)}\) in Eq. (71) or not
(see Table 3). Two different procedures are used in literature:
P1 Apply some resampling strategy to \(\{\boldsymbol {\mu }_{n,t-1}\}_{n=1}^{N}\), with probabilities according to the weights \(\rho _{n,t}^{(m)}\), to obtain \(\{\boldsymbol {\mu }_{n,t}\}_{n=1}^{N}\) [95, 274, 286]. Nonlinear transformations of \(\rho _{n,t}^{(m)}\) can also be applied [279].
P2 Build estimators of some moments of π employing \(\rho _{n,t}^{(m)}\), and use this information to obtain \(\{\boldsymbol {\mu }_{n,t}\}_{n=1}^{N}\) [96, 98, 287].