Particle filter

From a statistical and probabilistic viewpoint, particle filters belong to the class of branching/genetic type algorithms, and mean-field type interacting particle methodologies. The interpretation of these particle methods depends on the scientific discipline. In Evolutionary Computing, mean-field genetic type particle methodologies are often used as heuristic and natural search algorithms (a.k.a. Metaheuristic). In computational physics and molecular chemistry, they are used to solve Feynman-Kac path integration problems or to compute Boltzmann-Gibbs measures, top eigenvalues, and ground states of Schrödinger operators. In Biology and Genetics, they represent the evolution of a population of individuals or genes in some environment.

The origins of mean-field type evolutionary computational techniques can be traced back to 1950 and 1954 with Alan Turing's work on genetic type mutation-selection learning machines^[22] and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey.^[23][24] The first trace of particle filters in statistical methodology dates back to the mid-1950s; the 'Poor Man's Monte Carlo',^[25] that was proposed by John Hammersley et al., in 1954, contained hints of the genetic type particle filtering methods used today. In 1963, Nils Aall Barricelli simulated a genetic type algorithm to mimic the ability of individuals to play a simple game.^[26] In evolutionary computing literature, genetic-type mutation-selection algorithms became popular through the seminal work of John Holland in the early 1970s, particularly his book^[27] published in 1975.

In Biology and Genetics, the Australian geneticist Alex Fraser also published in 1957 a series of papers on the genetic type simulation of artificial selection of organisms.^[28] The computer simulation of the evolution by biologists became more common in the early 1960s, and the methods were described in books by Fraser and Burnell (1970)^[29] and Crosby (1973).^[30] Fraser's simulations included all of the essential elements of modern mutation-selection genetic particle algorithms.

From the mathematical viewpoint, the conditional distribution of the random states of a signal given some partial and noisy observations is described by a Feynman-Kac probability on the random trajectories of the signal weighted by a sequence of likelihood potential functions.^[7][8] Quantum Monte Carlo, and more specifically Diffusion Monte Carlo methods can also be interpreted as a mean-field genetic type particle approximation of Feynman-Kac path integrals.^{[7][8][9][13][14][31][32]} The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and Robert Richtmyer who developed in 1948 a mean-field particle interpretation of neutron chain reactions,^[33] but the first heuristic-like and genetic type particle algorithm (a.k.a. Resampled or Reconfiguration Monte Carlo methods) for estimating ground state energies of quantum systems (in reduced matrix models) is due to Jack H. Hetherington in 1984.^[13] One can also quote the earlier seminal works of Theodore E. Harris and Herman Kahn in particle physics, published in 1951, using mean-field but heuristic-like genetic methods for estimating particle transmission energies.^[34] In molecular chemistry, the use of genetic heuristic-like particle methodologies (a.k.a. pruning and enrichment strategies) can be traced back to 1955 with the seminal work of Marshall N. Rosenbluth and Arianna W. Rosenbluth.^[12]

The use of genetic particle algorithms in advanced signal processing and Bayesian inference is more recent. In January 1993, Genshiro Kitagawa developed a "Monte Carlo filter",^[35] a slightly modified version of this article appeared in 1996.^[36] In April 1993, Neil J. Gordon et al., published in their seminal work^[37] an application of genetic type algorithm in Bayesian statistical inference. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state space or the noise of the system. Independently, the ones by Pierre Del Moral^[2] and Himilcon Carvalho, Pierre Del Moral, André Monin, and Gérard Salut^[38] on particle filters published in the mid-1990s. Particle filters were also developed in signal processing in early 1989-1992 by P. Del Moral, J.C. Noyer, G. Rigal, and G. Salut in the LAAS-CNRS in a series of restricted and classified research reports with STCAN (Service Technique des Constructions et Armes Navales), the IT company DIGILOG, and the LAAS-CNRS (the Laboratory for Analysis and Architecture of Systems) on RADAR/SONAR and GPS signal processing problems.^{[39][40][41][42][43][44]}

From 1950 to 1996, all the publications on particle filters, and genetic algorithms, including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and genealogical and ancestral tree-based algorithms.

The mathematical foundations and the first rigorous analysis of these particle algorithms are due to Pierre Del Moral^[2][4] in 1996. The article^[2] also contains proof of the unbiased properties of a particle approximation of likelihood functions and unnormalized conditional probability measures. The unbiased particle estimator of the likelihood functions presented in this article is used today in Bayesian statistical inference.

Dan Crisan, Jessica Gaines, and Terry Lyons,^[45][46][47] as well as Pierre Del Moral, and Terry Lyons,^[48] created branching-type particle techniques with various population sizes around the end of the 1990s. P. Del Moral, A. Guionnet, and L. Miclo^[8][49][50] made more advances in this subject in 2000. Pierre Del Moral and Alice Guionnet^[51] proved the first central limit theorems in 1999, and Pierre Del Moral and Laurent Miclo^[8] proved them in 2000. The first uniform convergence results concerning the time parameter for particle filters were developed at the end of the 1990s by Pierre Del Moral and Alice Guionnet.^[49][50] The first rigorous analysis of genealogical tree-based particle filter smoothers is due to P. Del Moral and L. Miclo in 2001^[52]

The theory on Feynman-Kac particle methodologies and related particle filter algorithms was developed in 2000 and 2004 in the books.^[8][5] These abstract probabilistic models encapsulate genetic type algorithms, particle, and bootstrap filters, interacting Kalman filters (a.k.a. Rao–Blackwellized particle filter^[53]), importance sampling and resampling style particle filter techniques, including genealogical tree-based and particle backward methodologies for solving filtering and smoothing problems. Other classes of particle filtering methodologies include genealogical tree-based models,^[10][5][54] backward Markov particle models,^[10][55] adaptive mean-field particle models,^[6] island-type particle models,^[56][57] particle Markov chain Monte Carlo methodologies,^[58][59] Sequential Monte Carlo samplers ^[60][61][62] and Sequential Monte Carlo Approximate Bayesian Computation methods^[63] and Sequential Monte Carlo ABC based Bayesian Bootstrap.^[64]

A particle filter's goal is to estimate the posterior density of state variables given observation variables. The particle filter is intended for use with a hidden Markov Model, in which the system includes both hidden and observable variables. The observable variables (observation process) are linked to the hidden variables (state-process) via a known functional form. Similarly, the probabilistic description of the dynamical system defining the evolution of the state variables is known.

A generic particle filter estimates the posterior distribution of the hidden states using the observation measurement process. With respect to a state-space such as the one below:

${\displaystyle {\begin{array}{cccccccccc}X_{0}&\to &X_{1}&\to &X_{2}&\to &X_{3}&\to &\cdots &{\text{signal}}\\\downarrow &&\downarrow &&\downarrow &&\downarrow &&\cdots &\\Y_{0}&&Y_{1}&&Y_{2}&&Y_{3}&&\cdots &{\text{observation}}\end{array}}}$

the filtering problem is to estimate sequentially the values of the hidden states ${\displaystyle X_{k}}$, given the values of the observation process ${\displaystyle Y_{0},\cdots ,Y_{k},}$ at any time step k.

All Bayesian estimates of ${\displaystyle X_{k}}$ follow from the posterior density ${\displaystyle p(x_{k}|y_{0},y_{1},...,y_{k})}$. The particle filter methodology provides an approximation of these conditional probabilities using the empirical measure associated with a genetic type particle algorithm. In contrast, the Markov Chain Monte Carlo or importance sampling approach would model the full posterior ${\displaystyle p(x_{0},x_{1},...,x_{k}|y_{0},y_{1},...,y_{k})}$.

Particle methods often assume ${\displaystyle X_{k}}$ and the observations ${\displaystyle Y_{k}}$ can be modeled in this form:

• ${\displaystyle X_{0},X_{1},\cdots }$ is a Markov process on ${\displaystyle \mathbb {R} ^{d_{x}}}$ (for some ${\displaystyle d_{x}\geqslant 1}$) that evolves according to the transition probability density ${\displaystyle p(x_{k}|x_{k-1})}$. This model is also often written in a synthetic way as

  ${\displaystyle X_{k}|X_{k-1}=x_{k}\sim p(x_{k}|x_{k-1})}$

with an initial probability density ${\displaystyle p(x_{0})}$.

• The observations ${\displaystyle Y_{0},Y_{1},\cdots }$ take values in some state space on ${\displaystyle \mathbb {R} ^{d_{y}}}$ (for some ${\displaystyle d_{y}\geqslant 1}$) and are conditionally independent provided that ${\displaystyle X_{0},X_{1},\cdots }$ are known. In other words, each ${\displaystyle Y_{k}}$ only depends on ${\displaystyle X_{k}}$. In addition, we assume conditional distribution for ${\displaystyle Y_{k}}$ given ${\displaystyle X_{k}=x_{k}}$ are absolutely continuous, and in a synthetic way we have

  ${\displaystyle Y_{k}|X_{k}=y_{k}\sim p(y_{k}|x_{k})}$

An example of system with these properties is:

${\displaystyle X_{k}=g(X_{k-1})+W_{k-1}}$

${\displaystyle Y_{k}=h(X_{k})+V_{k}}$

where both ${\displaystyle W_{k}}$ and ${\displaystyle V_{k}}$ are mutually independent sequences with known probability density functions and g and h are known functions. These two equations can be viewed as state space equations and look similar to the state space equations for the Kalman filter. If the functions g and h in the above example are linear, and if both ${\displaystyle W_{k}}$ and ${\displaystyle V_{k}}$ are Gaussian, the Kalman filter finds the exact Bayesian filtering distribution. If not, Kalman filter-based methods are a first-order approximation (EKF) or a second-order approximation (UKF in general, but if the probability distribution is Gaussian a third-order approximation is possible).

The assumption that the initial distribution and the transitions of the Markov chain are continuous for the Lebesgue measure can be relaxed. To design a particle filter we simply need to assume that we can sample the transitions ${\displaystyle X_{k-1}\to X_{k}}$ of the Markov chain ${\displaystyle X_{k},}$ and to compute the likelihood function ${\displaystyle x_{k}\mapsto p(y_{k}|x_{k})}$ (see for instance the genetic selection mutation description of the particle filter given below). The continuous assumption on the Markov transitions of ${\displaystyle X_{k}}$ is only used to derive in an informal (and rather abusive) way different formulae between posterior distributions using the Bayes' rule for conditional densities.

In certain problems, the conditional distribution of observations, given the random states of the signal, may fail to have a density; the latter may be impossible or too complex to compute.^[19] In this situation, an additional level of approximation is necessitated. One strategy is to replace the signal ${\displaystyle X_{k}}$ by the Markov chain ${\displaystyle {\mathcal {X}}_{k}=\left(X_{k},Y_{k}\right)}$ and to introduce a virtual observation of the form

${\displaystyle {\mathcal {Y}}_{k}=Y_{k}+\epsilon {\mathcal {V}}_{k}\quad {\mbox{for some parameter}}\quad \epsilon \in [0,1]}$

for some sequence of independent random variables ${\displaystyle {\mathcal {V}}_{k}}$ with known probability density functions. The central idea is to observe that

${\displaystyle {\text{Law}}\left(X_{k}|{\mathcal {Y}}_{0}=y_{0},\cdots ,{\mathcal {Y}}_{k}=y_{k}\right)\approx _{\epsilon \downarrow 0}{\text{Law}}\left(X_{k}|Y_{0}=y_{0},\cdots ,Y_{k}=y_{k}\right)}$

The particle filter associated with the Markov process ${\displaystyle {\mathcal {X}}_{k}=\left(X_{k},Y_{k}\right)}$ given the partial observations ${\displaystyle {\mathcal {Y}}_{0}=y_{0},\cdots ,{\mathcal {Y}}_{k}=y_{k},}$ is defined in terms of particles evolving in ${\displaystyle \mathbb {R} ^{d_{x}+d_{y}}}$ with a likelihood function given with some obvious abusive notation by ${\displaystyle p({\mathcal {Y}}_{k}|{\mathcal {X}}_{k})}$. These probabilistic techniques are closely related to Approximate Bayesian Computation (ABC). In the context of particle filters, these ABC particle filtering techniques were introduced in 1998 by P. Del Moral, J. Jacod and P. Protter.^[65] They were further developed by P. Del Moral, A. Doucet and A. Jasra.^[66][67]

Bayes' rule for conditional probability gives:

${\displaystyle p(x_{0},\cdots ,x_{k}|y_{0},\cdots ,y_{k})={\frac {p(y_{0},\cdots ,y_{k}|x_{0},\cdots ,x_{k})p(x_{0},\cdots ,x_{k})}{p(y_{0},\cdots ,y_{k})}}}$

where

${\displaystyle {\begin{aligned}p(y_{0},\cdots ,y_{k})&=\int p(y_{0},\cdots ,y_{k}|x_{0},\cdots ,x_{k})p(x_{0},\cdots ,x_{k})dx_{0}\cdots dx_{k}\\p(y_{0},\cdots ,y_{k}|x_{0},\cdots ,x_{k})&=\prod _{l=0}^{k}p(y_{l}|x_{l})\\p(x_{0},\cdots ,x_{k})&=p_{0}(x_{0})\prod _{l=1}^{k}p(x_{l}|x_{l-1})\end{aligned}}}$

Particle filters are also an approximation, but with enough particles they can be much more accurate.^{[2][4][5][49][50]} The nonlinear filtering equation is given by the recursion

with the convention ${\displaystyle p(x_{0}|y_{0},\cdots ,y_{k-1})=p(x_{0})}$ for k = 0. The nonlinear filtering problem consists in computing these conditional distributions sequentially.

We fix a time horizon n and a sequence of observations ${\displaystyle Y_{0}=y_{0},\cdots ,Y_{n}=y_{n}}$, and for each k = 0, ..., n we set:

${\displaystyle G_{k}(x_{k})=p(y_{k}|x_{k}).}$

In this notation, for any bounded function F on the set of trajectories of ${\displaystyle X_{k}}$ from the origin k = 0 up to time k = n, we have the Feynman-Kac formula

${\displaystyle {\begin{aligned}\int F(x_{0},\cdots ,x_{n})p(x_{0},\cdots ,x_{n}|y_{0},\cdots ,y_{n})dx_{0}\cdots dx_{n}&={\frac {\int F(x_{0},\cdots ,x_{n})\left\{\prod \limits _{k=0}^{n}p(y_{k}|x_{k})\right\}p(x_{0},\cdots ,x_{n})dx_{0}\cdots dx_{n}}{\int \left\{\prod \limits _{k=0}^{n}p(y_{k}|x_{k})\right\}p(x_{0},\cdots ,x_{n})dx_{0}\cdots dx_{n}}}\\&={\frac {E\left(F(X_{0},\cdots ,X_{n})\prod \limits _{k=0}^{n}G_{k}(X_{k})\right)}{E\left(\prod \limits _{k=0}^{n}G_{k}(X_{k})\right)}}\end{aligned}}}$

Feynman-Kac path integration models arise in a variety of scientific disciplines, including in computational physics, biology, information theory and computer sciences.^[8][10][5] Their interpretations are dependent on the application domain. For instance, if we choose the indicator function ${\displaystyle G_{n}(x_{n})=1_{A}(x_{n})}$ of some subset of the state space, they represent the conditional distribution of a Markov chain given it stays in a given tube; that is, we have:

${\displaystyle E\left(F(X_{0},\cdots ,X_{n})|X_{0}\in A,\cdots ,X_{n}\in A\right)={\frac {E\left(F(X_{0},\cdots ,X_{n})\prod \limits _{k=0}^{n}G_{k}(X_{k})\right)}{E\left(\prod \limits _{k=0}^{n}G_{k}(X_{k})\right)}}}$

and

${\displaystyle P\left(X_{0}\in A,\cdots ,X_{n}\in A\right)=E\left(\prod \limits _{k=0}^{n}G_{k}(X_{k})\right)}$

as soon as the normalizing constant is strictly positive.

Initially, such an algorithm starts with N independent random variables ${\displaystyle \left(\xi _{0}^{i}\right)_{1\leqslant i\leqslant N}}$ with common probability density ${\displaystyle p(x_{0})}$. The genetic algorithm selection-mutation transitions^[2][4]

${\displaystyle \xi _{k}:=\left(\xi _{k}^{i}\right)_{1\leqslant i\leqslant N}{\stackrel {\text{selection}}{\longrightarrow }}{\widehat {\xi }}_{k}:=\left({\widehat {\xi }}_{k}^{i}\right)_{1\leqslant i\leqslant N}{\stackrel {\text{mutation}}{\longrightarrow }}\xi _{k+1}:=\left(\xi _{k+1}^{i}\right)_{1\leqslant i\leqslant N}}$

mimic/approximate the updating-prediction transitions of the optimal filter evolution (Eq. 1):

• During the selection-updating transition we sample N (conditionally) independent random variables ${\displaystyle {\widehat {\xi }}_{k}:=\left({\widehat {\xi }}_{k}^{i}\right)_{1\leqslant i\leqslant N}}$ with common (conditional) distribution

${\displaystyle \sum _{i=1}^{N}{\frac {p(y_{k}|\xi _{k}^{i})}{\sum _{j=1}^{N}p(y_{k}|\xi _{k}^{j})}}\delta _{\xi _{k}^{i}}(dx_{k})}$

where ${\displaystyle \delta _{a}}$ stands for the Dirac measure at a given state a.

• During the mutation-prediction transition, from each selected particle ${\displaystyle {\widehat {\xi }}_{k}^{i}}$ we sample independently a transition

${\displaystyle {\widehat {\xi }}_{k}^{i}\longrightarrow \xi _{k+1}^{i}\sim p(x_{k+1}|{\widehat {\xi }}_{k}^{i}),\qquad i=1,\cdots ,N.}$

In the above displayed formulae ${\displaystyle p(y_{k}|\xi _{k}^{i})}$ stands for the likelihood function ${\displaystyle x_{k}\mapsto p(y_{k}|x_{k})}$ evaluated at ${\displaystyle x_{k}=\xi _{k}^{i}}$, and ${\displaystyle p(x_{k+1}|{\widehat {\xi }}_{k}^{i})}$ stands for the conditional density ${\displaystyle p(x_{k+1}|x_{k})}$ evaluated at ${\displaystyle x_{k}={\widehat {\xi }}_{k}^{i}}$.

At each time k, we have the particle approximations

${\displaystyle {\widehat {p}}(dx_{k}|y_{0},\cdots ,y_{k}):={\frac {1}{N}}\sum _{i=1}^{N}\delta _{{\widehat {\xi }}_{k}^{i}}(dx_{k})\approx _{N\uparrow \infty }p(dx_{k}|y_{0},\cdots ,y_{k})\approx _{N\uparrow \infty }\sum _{i=1}^{N}{\frac {p(y_{k}|\xi _{k}^{i})}{\sum _{i=1}^{N}p(y_{k}|\xi _{k}^{j})}}\delta _{\xi _{k}^{i}}(dx_{k})}$

and

${\displaystyle {\widehat {p}}(dx_{k}|y_{0},\cdots ,y_{k-1}):={\frac {1}{N}}\sum _{i=1}^{N}\delta _{\xi _{k}^{i}}(dx_{k})\approx _{N\uparrow \infty }p(dx_{k}|y_{0},\cdots ,y_{k-1})}$

In Genetic algorithms and Evolutionary computing community, the mutation-selection Markov chain described above is often called the genetic algorithm with proportional selection. Several branching variants, including with random population sizes have also been proposed in the articles.^[5][45][48]

Particle methods, like all sampling-based approaches (e.g., Markov Chain Monte Carlo), generate a set of samples that approximate the filtering density

${\displaystyle p(x_{k}|y_{0},\cdots ,y_{k}).}$

For example, we may have N samples from the approximate posterior distribution of ${\displaystyle X_{k}}$, where the samples are labeled with superscripts as:

${\displaystyle {\widehat {\xi }}_{k}^{1},\cdots ,{\widehat {\xi }}_{k}^{N}.}$

Then, expectations with respect to the filtering distribution are approximated by

${\displaystyle \int f(x_{k})p(x_{k}|y_{0},\cdots ,y_{k})\,dx_{k}\approx _{N\uparrow \infty }{\frac {1}{N}}\sum _{i=1}^{N}f\left({\widehat {\xi }}_{k}^{i}\right)=\int f(x_{k}){\widehat {p}}(dx_{k}|y_{0},\cdots ,y_{k})}$

Eq. 2

with

${\displaystyle {\widehat {p}}(dx_{k}|y_{0},\cdots ,y_{k})={\frac {1}{N}}\sum _{i=1}^{N}\delta _{{\widehat {\xi }}_{k}^{i}}(dx_{k})}$

where ${\displaystyle \delta _{a}}$ stands for the Dirac measure at a given state a. The function f, in the usual way for Monte Carlo, can give all the moments etc. of the distribution up to some approximation error. When the approximation equation (Eq. 2) is satisfied for any bounded function f we write

${\displaystyle p(dx_{k}|y_{0},\cdots ,y_{k}):=p(x_{k}|y_{0},\cdots ,y_{k})dx_{k}\approx _{N\uparrow \infty }{\widehat {p}}(dx_{k}|y_{0},\cdots ,y_{k})={\frac {1}{N}}\sum _{i=1}^{N}\delta _{{\widehat {\xi }}_{k}^{i}}(dx_{k})}$

Particle filters can be interpreted as a genetic type particle algorithm evolving with mutation and selection transitions. We can keep track of the ancestral lines

${\displaystyle \left({\widehat {\xi }}_{0,k}^{i},{\widehat {\xi }}_{1,k}^{i},\cdots ,{\widehat {\xi }}_{k-1,k}^{i},{\widehat {\xi }}_{k,k}^{i}\right)}$

of the particles ${\displaystyle i=1,\cdots ,N}$. The random states ${\displaystyle {\widehat {\xi }}_{l,k}^{i}}$, with the lower indices l=0,...,k, stands for the ancestor of the individual ${\displaystyle {\widehat {\xi }}_{k,k}^{i}={\widehat {\xi }}_{k}^{i}}$ at level l=0,...,k. In this situation, we have the approximation formula

${\displaystyle {\begin{aligned}\int F(x_{0},\cdots ,x_{k})p(x_{0},\cdots ,x_{k}|y_{0},\cdots ,y_{k})\,dx_{0}\cdots dx_{k}&\approx _{N\uparrow \infty }{\frac {1}{N}}\sum _{i=1}^{N}F\left({\widehat {\xi }}_{0,k}^{i},{\widehat {\xi }}_{1,k}^{i},\cdots ,{\widehat {\xi }}_{k,k}^{i}\right)\\&=\int F(x_{0},\cdots ,x_{k}){\widehat {p}}(d(x_{0},\cdots ,x_{k})|y_{0},\cdots ,y_{k})\end{aligned}}}$

Eq. 3

with the empirical measure

${\displaystyle {\widehat {p}}(d(x_{0},\cdots ,x_{k})|y_{0},\cdots ,y_{k}):={\frac {1}{N}}\sum _{i=1}^{N}\delta _{\left({\widehat {\xi }}_{0,k}^{i},{\widehat {\xi }}_{1,k}^{i},\cdots ,{\widehat {\xi }}_{k,k}^{i}\right)}(d(x_{0},\cdots ,x_{k}))}$

Here F stands for any founded function on the path space of the signal. In a more synthetic form (Eq. 3) is equivalent to

${\displaystyle {\begin{aligned}p(d(x_{0},\cdots ,x_{k})|y_{0},\cdots ,y_{k})&:=p(x_{0},\cdots ,x_{k}|y_{0},\cdots ,y_{k})\,dx_{0}\cdots dx_{k}\\&\approx _{N\uparrow \infty }{\widehat {p}}(d(x_{0},\cdots ,x_{k})|y_{0},\cdots ,y_{k})\\&:={\frac {1}{N}}\sum _{i=1}^{N}\delta _{\left({\widehat {\xi }}_{0,k}^{i},\cdots ,{\widehat {\xi }}_{k,k}^{i}\right)}(d(x_{0},\cdots ,x_{k}))\end{aligned}}}$

Particle filters can be interpreted in many different ways. From the probabilistic point of view they coincide with a mean-field particle interpretation of the nonlinear filtering equation. The updating-prediction transitions of the optimal filter evolution can also be interpreted as the classical genetic type selection-mutation transitions of individuals. The sequential importance resampling technique provides another interpretation of the filtering transitions coupling importance sampling with the bootstrap resampling step. Last, but not least, particle filters can be seen as an acceptance-rejection methodology equipped with a recycling mechanism.^[10][5]

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The nonlinear filtering evolution can be interpreted as a dynamical system in the set of probability measures of the form ${\displaystyle \eta _{n+1}=\Phi _{n+1}\left(\eta _{n}\right)}$ where ${\displaystyle \Phi _{n+1}}$ stands for some mapping from the set of probability distribution into itself. For instance, the evolution of the one-step optimal predictor ${\displaystyle \eta _{n}(dx_{n})=p(x_{n}|y_{0},\cdots ,y_{n-1})dx_{n}}$

satisfies a nonlinear evolution starting with the probability distribution ${\displaystyle \eta _{0}(dx_{0})=p(x_{0})dx_{0}}$. One of the simplest ways to approximate these probability measures is to start with N independent random variables ${\displaystyle \left(\xi _{0}^{i}\right)_{1\leqslant i\leqslant N}}$ with common probability distribution ${\displaystyle \eta _{0}(dx_{0})=p(x_{0})dx_{0}}$ . Suppose we have defined a sequence of N random variables ${\displaystyle \left(\xi _{n}^{i}\right)_{1\leqslant i\leqslant N}}$ such that

${\displaystyle {\frac {1}{N}}\sum _{i=1}^{N}\delta _{\xi _{n}^{i}}(dx_{n})\approx _{N\uparrow \infty }\eta _{n}(dx_{n})}$

At the next step we sample N (conditionally) independent random variables ${\displaystyle \xi _{n+1}:=\left(\xi _{n+1}^{i}\right)_{1\leqslant i\leqslant N}}$ with common law .

${\displaystyle \Phi _{n+1}\left({\frac {1}{N}}\sum _{i=1}^{N}\delta _{\xi _{n}^{i}}\right)\approx _{N\uparrow \infty }\Phi _{n+1}\left(\eta _{n}\right)=\eta _{n+1}}$

We illustrate this mean-field particle principle in the context of the evolution of the one step optimal predictors

For k = 0 we use the convention ${\displaystyle p(x_{0}|y_{0},\cdots ,y_{-1}):=p(x_{0})}$.

By the law of large numbers, we have

${\displaystyle {\widehat {p}}(dx_{0})={\frac {1}{N}}\sum _{i=1}^{N}\delta _{\xi _{0}^{i}}(dx_{0})\approx _{N\uparrow \infty }p(x_{0})dx_{0}}$

in the sense that

${\displaystyle \int f(x_{0}){\widehat {p}}(dx_{0})={\frac {1}{N}}\sum _{i=1}^{N}f(\xi _{0}^{i})\approx _{N\uparrow \infty }\int f(x_{0})p(dx_{0})dx_{0}}$

for any bounded function ${\displaystyle f}$. We further assume that we have constructed a sequence of particles ${\displaystyle \left(\xi _{k}^{i}\right)_{1\leqslant i\leqslant N}}$ at some rank k such that

${\displaystyle {\widehat {p}}(dx_{k}|y_{0},\cdots ,y_{k-1}):={\frac {1}{N}}\sum _{i=1}^{N}\delta _{\xi _{k}^{i}}(dx_{k})\approx _{N\uparrow \infty }~p(x_{k}~|~y_{0},\cdots ,y_{k-1})dx_{k}}$

in the sense that for any bounded function ${\displaystyle f}$ we have

${\displaystyle \int f(x_{k}){\widehat {p}}(dx_{k}|y_{0},\cdots ,y_{k-1})={\frac {1}{N}}\sum _{i=1}^{N}f(\xi _{k}^{i})\approx _{N\uparrow \infty }\int f(x_{k})p(dx_{k}|y_{0},\cdots ,y_{k-1})dx_{k}}$

In this situation, replacing ${\displaystyle p(x_{k}|y_{0},\cdots ,y_{k-1})dx_{k}}$ by the empirical measure ${\displaystyle {\widehat {p}}(dx_{k}|y_{0},\cdots ,y_{k-1})}$ in the evolution equation of the one-step optimal filter stated in (Eq. 4) we find that

${\displaystyle p(x_{k+1}|y_{0},\cdots ,y_{k})\approx _{N\uparrow \infty }\int p(x_{k+1}|x'_{k}){\frac {p(y_{k}|x_{k}'){\widehat {p}}(dx'_{k}|y_{0},\cdots ,y_{k-1})}{\int p(y_{k}|x''_{k}){\widehat {p}}(dx''_{k}|y_{0},\cdots ,y_{k-1})}}}$

Notice that the right hand side in the above formula is a weighted probability mixture

${\displaystyle \int p(x_{k+1}|x'_{k}){\frac {p(y_{k}|x_{k}'){\widehat {p}}(dx'_{k}|y_{0},\cdots ,y_{k-1})}{\int p(y_{k}|x''_{k}){\widehat {p}}(dx''_{k}|y_{0},\cdots ,y_{k-1})}}=\sum _{i=1}^{N}{\frac {p(y_{k}|\xi _{k}^{i})}{\sum _{i=1}^{N}p(y_{k}|\xi _{k}^{j})}}p(x_{k+1}|\xi _{k}^{i})=:{\widehat {q}}(x_{k+1}|y_{0},\cdots ,y_{k})}$

where ${\displaystyle p(y_{k}|\xi _{k}^{i})}$ stands for the density ${\displaystyle p(y_{k}|x_{k})}$ evaluated at ${\displaystyle x_{k}=\xi _{k}^{i}}$, and ${\displaystyle p(x_{k+1}|\xi _{k}^{i})}$ stands for the density ${\displaystyle p(x_{k+1}|x_{k})}$ evaluated at ${\displaystyle x_{k}=\xi _{k}^{i}}$ for ${\displaystyle i=1,\cdots ,N.}$

Then, we sample N independent random variable ${\displaystyle \left(\xi _{k+1}^{i}\right)_{1\leqslant i\leqslant N}}$ with common probability density ${\displaystyle {\widehat {q}}(x_{k+1}|y_{0},\cdots ,y_{k})}$ so that

${\displaystyle {\widehat {p}}(dx_{k+1}|y_{0},\cdots ,y_{k}):={\frac {1}{N}}\sum _{i=1}^{N}\delta _{\xi _{k+1}^{i}}(dx_{k+1})\approx _{N\uparrow \infty }{\widehat {q}}(x_{k+1}|y_{0},\cdots ,y_{k})dx_{k+1}\approx _{N\uparrow \infty }p(x_{k+1}|y_{0},\cdots ,y_{k})dx_{k+1}}$

Iterating this procedure, we design a Markov chain such that

${\displaystyle {\widehat {p}}(dx_{k}|y_{0},\cdots ,y_{k-1}):={\frac {1}{N}}\sum _{i=1}^{N}\delta _{\xi _{k}^{i}}(dx_{k})\approx _{N\uparrow \infty }p(dx_{k}|y_{0},\cdots ,y_{k-1}):=p(x_{k}|y_{0},\cdots ,y_{k-1})dx_{k}}$

Notice that the optimal filter is approximated at each time step k using the Bayes' formulae

${\displaystyle p(dx_{k}|y_{0},\cdots ,y_{k})\approx _{N\uparrow \infty }{\frac {p(y_{k}|x_{k}){\widehat {p}}(dx_{k}|y_{0},\cdots ,y_{k-1})}{\int p(y_{k}|x'_{k}){\widehat {p}}(dx'_{k}|y_{0},\cdots ,y_{k-1})}}=\sum _{i=1}^{N}{\frac {p(y_{k}|\xi _{k}^{i})}{\sum _{j=1}^{N}p(y_{k}|\xi _{k}^{j})}}~\delta _{\xi _{k}^{i}}(dx_{k})}$

The terminology "mean-field approximation" comes from the fact that we replace at each time step the probability measure ${\displaystyle p(dx_{k}|y_{0},\cdots ,y_{k-1})}$ by the empirical approximation ${\displaystyle {\widehat {p}}(dx_{k}|y_{0},\cdots ,y_{k-1})}$. The mean-field particle approximation of the filtering problem is far from being unique. Several strategies are developed in the books.^[10][5]

The analysis of the convergence of particle filters was started in 1996^[2][4] and in 2000 in the book^[8] and the series of articles.^{[48][49][50][51][52][68][69]} More recent developments can be found in the books,^[10][5] When the filtering equation is stable (in the sense that it corrects any erroneous initial condition), the bias and the variance of the particle particle estimates

${\displaystyle I_{k}(f):=\int f(x_{k})p(dx_{k}|y_{0},\cdots ,y_{k-1})\approx _{N\uparrow \infty }{\widehat {I}}_{k}(f):=\int f(x_{k}){\widehat {p}}(dx_{k}|y_{0},\cdots ,y_{k-1})}$

are controlled by the non asymptotic uniform estimates

${\displaystyle \sup _{k\geqslant 0}\left\vert E\left({\widehat {I}}_{k}(f)\right)-I_{k}(f)\right\vert \leqslant {\frac {c_{1}}{N}}}$

${\displaystyle \sup _{k\geqslant 0}E\left(\left[{\widehat {I}}_{k}(f)-I_{k}(f)\right]^{2}\right)\leqslant {\frac {c_{2}}{N}}}$

for any function f bounded by 1, and for some finite constants ${\displaystyle c_{1},c_{2}.}$ In addition, for any ${\displaystyle x\geqslant 0}$:

${\displaystyle \mathbf {P} \left(\left|{\widehat {I}}_{k}(f)-I_{k}(f)\right|\leqslant c_{1}{\frac {x}{N}}+c_{2}{\sqrt {\frac {x}{N}}}\land \sup _{0\leqslant k\leqslant n}\left|{\widehat {I}}_{k}(f)-I_{k}(f)\right|\leqslant c{\sqrt {\frac {x\log(n)}{N}}}\right)>1-e^{-x}}$

for some finite constants ${\displaystyle c_{1},c_{2}}$ related to the asymptotic bias and variance of the particle estimate, and some finite constant c. The same results are satisfied if we replace the one step optimal predictor by the optimal filter approximation.

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Tracing back in time the ancestral lines

${\displaystyle \left({\widehat {\xi }}_{0,k}^{i},{\widehat {\xi }}_{1,k}^{i},\cdots ,{\widehat {\xi }}_{k-1,k}^{i},{\widehat {\xi }}_{k,k}^{i}\right),\quad \left(\xi _{0,k}^{i},\xi _{1,k}^{i},\cdots ,\xi _{k-1,k}^{i},\xi _{k,k}^{i}\right)}$

of the individuals ${\displaystyle {\widehat {\xi }}_{k}^{i}\left(={\widehat {\xi }}_{k,k}^{i}\right)}$ and ${\displaystyle \xi _{k}^{i}\left(={\xi }_{k,k}^{i}\right)}$ at every time step k, we also have the particle approximations

${\displaystyle {\begin{aligned}{\widehat {p}}(d(x_{0},\cdots ,x_{k})|y_{0},\cdots ,y_{k})&:={\frac {1}{N}}\sum _{i=1}^{N}\delta _{\left({\widehat {\xi }}_{0,k}^{i},\cdots ,{\widehat {\xi }}_{0,k}^{i}\right)}(d(x_{0},\cdots ,x_{k}))\\&\approx _{N\uparrow \infty }p(d(x_{0},\cdots ,x_{k})|y_{0},\cdots ,y_{k})\\&\approx _{N\uparrow \infty }\sum _{i=1}^{N}{\frac {p(y_{k}|\xi _{k,k}^{i})}{\sum _{j=1}^{N}p(y_{k}|\xi _{k,k}^{j})}}\delta _{\left(\xi _{0,k}^{i},\cdots ,\xi _{0,k}^{i}\right)}(d(x_{0},\cdots ,x_{k}))\\&\ \\{\widehat {p}}(d(x_{0},\cdots ,x_{k})|y_{0},\cdots ,y_{k-1})&:={\frac {1}{N}}\sum _{i=1}^{N}\delta _{\left(\xi _{0,k}^{i},\cdots ,\xi _{k,k}^{i}\right)}(d(x_{0},\cdots ,x_{k}))\\&\approx _{N\uparrow \infty }p(d(x_{0},\cdots ,x_{k})|y_{0},\cdots ,y_{k-1})\\&:=p(x_{0},\cdots ,x_{k}|y_{0},\cdots ,y_{k-1})dx_{0},\cdots ,dx_{k}\end{aligned}}}$