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Table 1 Particle filtering using SIS

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

\(\left [\left \{x^{i}_{n}, w^{i}_{n}\right \}^{M}_{i=1}\right ]\) = SIS\(\,\,\left [\left \{ x^{i}_{n-1}, w^{i}_{n-1}\right \}^{M}_{i=1},\,y_{n}\right ]\)
– 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 \(x^{i}_{n}\sim q\left (x_{n}\mid x^{i}_{n-1},\, y_{1:n}\right)\), for i=1,2,,M.
– Calculate importance weights as
\( {w}^{i}_{n}= w^{i}_{n-1} \frac {{p}\left (y_{n}\mid x^{i}_{n}\right)\,{p}\left (x^{i}_{n}\mid {x^{i}_{n-1}}\right)}{q\left (x^{i}_{n} \mid {x^{i}_{n-1},\,y_{1:n}}\right)} \)
– Normalize : \(W^{i}_{n} = w^{i}_{n}\,/\,{\sum ^{M}_{i=1}{w}^{i}_{n}}\)
– Propagate : \(\left \{x^{i}_{n}, w^{i}_{n}\right \}^{M}_{i=1}\)