# Particle filters

incorporating Interacting Particle Systems, Sequential Monte Carlo and a profusion of other simultaneous-discovery names

July 25, 2014 — March 24, 2023

A Monte Carlo algorithm which updates a population of samples with a nested update. The easiest entry point is IMO to think about random-sample generalisation of state filter models via importance sampling. These are classically considered cousins to the linear Gaussian Kalman filter, applicable to more challenging models at the cost of using Monte Carlo approximations.

This has nothing to do with filters for particulate matter as seen in respirators.

There is too much confusing and unhelpful terminology here, and I am only at the fringe of this field so I will not attempt to typologize. Let us clear up the main stumbling block though: somehow the theoretical basis field has coalesced under the banner of *interacting particle systems* which is an awful unsearchable name which could mean anything, and indeed does in other disciplines. Wikipedia disambiguates this problem with the gritty and also abstruse Mean Field Particle Methods. In practical applications we talk about *particle filters*, or *sequential Monte Carlo*, or *bootstrap filters*, or *iterated importance sampling* and these all mean confusingly similar things.

## 1 Introductions

- Pierre Jacob’s Particle methods for statistics reading list
- The lineage and reasoning is well explained by Cappe, Godsill, and Moulines (2007).
- Chopin and Papaspiliopoulos (2020)

Easy to explain with an example such as this particle filter in scala.

## 2 Feynman-Kac formulae

See Feynman-Kac.

## 3 System Identification in

Do not know the parameters governing the system dynamics and need to learn those too? See System identification with particle fitlers.

## 4 Relation to Ensemble Kalman filters

Yes.

## 5 Particle flow

Introduced to me by my colleague Adrian Bishop Profile.

Bunch and Godsill (2014); Daum, Huang, and Noushin (2010); Daum and Huang (2010); Daum and Huang (2009); Daum and Huang (2008); Daum and Huang (2013)

## 6 Non-Gaussian evolution

### 6.1 Jump process

I am interested in jump-process SMC (to be defined). For those I should apparently consult Graham and Méléard (1997);Grünbaum (1971);Méléard (1996);Shiga and Tanaka (1985).

### 6.2 On weird graphs

Christian Andersson Naesseth, Lindsten, and Schön (2014)

## 7 Rao-Blackwellized particle filter

Particles which represent marginalised densities (Murphy 2012, 23.6).

## 8 Tooling

Some MCMC toolkits incorporate SMC too.

- particles is a python library for teaching DIY particle filtering, to accompany the book Chopin and Papaspiliopoulos (2020).
- Johansen’s page, with C++ software
- Dirk Eddelbuettel’s lab created RCppSMC for R integration of the Johansen stuff. Documentation is not great — it only consists of black-box toy problems without any hint of how you would construct, e.g. a likelihood function, so I can’t evaluate how easy this would be to use, as opp plain C++.
- most probabilistic programming languages include a particle filter example.

## 9 References

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*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.

*IEEE Transactions on Signal Processing*.

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*arXiv:2102.07850 [Cs, Stat]*.

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*Journal of Econometrics*, Semiparametric methods in econometrics,.

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*The Annals of Applied Probability*.

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*arXiv:2007.02692 [q-Fin]*.