Auxiliary particle filter

The auxiliary particle filter is a particle filtering algorithm was introduced by Pitt and Shephard in 1999 to improve some deficiencies of the SIR algorithm when dealing with tailed observation densities.

Assume that the filtered posterior is described by the following "M" weighted samples:

: $p(x\_t|z\_\{1:t\})\; approx\; sum\_\{i=1\}^M\; omega^\{(i)\}\_t\; delta\; left(\; x\_t\; -\; x^\{(i)\}\_t\; ight).$

Then, each step in the algorithm consists of first drawing a sample of the particle index $k$ which will be propragated from $t-1$ into the new step $t$. These indexes are auxiliary variables only used as an intermediary step, hence the name of the algorithm. The indexes are drawn according to the likelihood of some reference point $mu^\{(i)\}\_t$ which in some way is related to the transition model $x\_t|x\_\{t-1\}$ (for example, the mean, a sample, etc.):

: $k^\{(i)\}\; sim\; P(i=k|z\_t)\; propto\; omega^\{(i)\}\_t\; p(\; z\_t\; |\; mu^\{(i)\}\_t\; )$

This is repeated for $i=1,2,dots,M$, and using these indexes we can now draw the conditional samples:

: $x\_t^\{(i)\}\; sim\; p(\; x\_t\; |\; x^\{k^\{(i)\_\{t-1\}).$

Finally, the weights are updated to account for the mismatch between the likelihood at the actual sample and the predicted point $mu\_t^\{k^\{(i)$:

: $omega\_t^\{(i)\}\; propto\; frac\{p(\; z\_t\; |\; x^\{(i)\}\_t)\; \}\; \{\; p(\; z\_t\; |\; mu^\{k^\{(i)\_t)\; \}.$

References

* cite journal
author = Pitt, M.K.
coauthors = Shephard, N.
year = 1999
title = Filtering Via Simulation: Auxiliary Particle Filters
journal = Journal of the American Statistical Association
volume = 94
issue = 446
pages = 590–591
url = http://www.questia.com/PM.qst?a=o&se=gglsc&d=5002321997
accessdate = 2008-05-06
doi = 10.2307/2670179