Feynman-Kac formulae

January 27, 2021 — January 27, 2021

Bayes
Monte Carlo
probabilistic algorithms
probability
state space models
statistics
swarm
time series
Figure 1

There is a mathematically rich theory about sequential Monte Carlo filters, and the central tool to make that go seems to be Feynman-Kac formulae. I don’t know if Feynman or Kac has much to do with raising this method, but their offspring seems to be something like “the central limit theorem for SMC”.

The notoriously abstruse Del Moral (2004) and Doucet, Freitas, and Gordon (2001) are regarded as the unifying introductions to these formulae, whatever they are. Diligent study will supposedly make consistent the diffusion processes and Feynman-Kac formulae and “propagation of chaos” and their delicate relationships. I will get around to understanding them myself eventually, maybe?

Cheng and Reich (2014) translates the Del Moral (French probabilist?) terminology into my more workaday statistician’s language.

Related, apparently: Backward SDEs.

1 References

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Cappé, Godsill, and Moulines. 2007. “An Overview of Existing Methods and Recent Advances in Sequential Monte Carlo.” Proceedings of the IEEE.
Cérou, Moral, Furon, et al. 2011. “Sequential Monte Carlo for Rare Event Estimation.” Statistics and Computing.
Chan-Wai-Nam, Mikael, and Warin. 2019. “Machine Learning for Semi Linear PDEs.” Journal of Scientific Computing.
Cheng, and Reich. 2014. “A McKean Optimal Transportation Perspective on Feynman-Kac Formulae with Application to Data Assimilation.”
Chopin, and Papaspiliopoulos. 2020. An Introduction to Sequential Monte Carlo. Springer Series in Statistics.
Chuang. 2010. “The Feynman-Kac Formula: Relationships Between Stochastic Differential Equations and Partial Differential Equations.”
Del Moral. 2004. Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications.
Del Moral, and Doucet. 2010. “Interacting Markov Chain Monte Carlo Methods for Solving Nonlinear Measure-Valued Equations.” The Annals of Applied Probability.
Del Moral, Hu, and Wu. 2011. On the Concentration Properties of Interacting Particle Processes.
Del Moral, and Miclo. 2000. “Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering.” In Séminaire de Probabilités XXXIV. Lecture Notes in Mathematics 1729.
Doucet, Freitas, and Gordon. 2001. Sequential Monte Carlo Methods in Practice.
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