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Table 2 Particle filtering using SIR

From: A weighted likelihood criteria for learning importance densities in particle filtering

\(\left [\left \{x^{i}_{n}, 1/M\right \}^{M}_{i=1}\right ]\) = SIR\(\,\,\left [\left \{ x^{i}_{n-1}, 1/M \right \}^{M}_{i=1},\,y_{n}\right ]\)
–Initialize \(x_{0}^{i} \sim p_{0}\) and \(w_{0}^{i} = 1/M\) for i=1,2,,M.
–DO for n=1,2,,T:
– Draw \(x^{*i}_{n}\sim q(x_{n}\mid x^{i}_{n-1}, y_{1:n})\), for i=1,2,,M.
– Calculate importance weights as
\({w}^{*i}_{n}= \frac {{p}\left (y_{n}\mid x^{i}_{n}\right){p}\left (x^{i}_{n}\mid { x^{i}_{n-1}}\right)}{q\left (x^{i}_{n}\mid { x^{i}_{n-1},y_{1:n}}\right)}\)
– Normalize : \(W^{*i}_{n} = w^{*i}_{n}\,/\,{\sum ^{M}_{i=1}{w}^{*i}_{n}}\)
– Resample \(\left \{x^{i}_{n}\right \}_{i=1}^{M}\) from \(\left \{x^{*i}_{n}\right \}_{i=1}^{M}\) with weights \(\left \{\,W^{*i}_{n}\,\right \}_{i=1}^{M}\)
– Propagate : \(\left \{x^{i}_{n}, 1/M\right \}^{M}_{i=1}\)