Importance sampling: Difference between revisions

Revision as of 19:30, 3 September 2024

Revision as of 13:19, 29 October 2024

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{{short description|Distribution estimation technique}}

'''Importance sampling''' is a [[Monte Carlo method]] for evaluating properties of a particular [[probability distribution|distribution]], while only having samples generated from a different distribution than the distribution of interest. Its introduction in statistics is generally attributed to a paper by [[Teun Kloek]] and [[Herman K. van Dijk]] in 1978,<ref>{{cite journal |first1=T. |last1=Kloek |first2=H. K. |last2=van Dijk |title=Bayesian Estimates of Equation System Parameters: An Application of Integration by Monte Carlo |journal=[[Econometrica]] |volume=46 |issue=1 |year=1978 |pages=1–19 |doi=10.2307/1913641 |jstor=1913641 |url=https://ageconsearch.umn.edu/record/272139/files/erasmus076.pdf }}</ref> but its precursors can be found in [[Monte Carlo method in statistical physics|statistical physics]] as early as 1949.<ref>{{cite journal |first=G. |last=Goertzle |authorlink=Gerald Goertzel |title=Quota Sampling and Importance Functions in Stochastic Solution of Particle Problems |journal=Technical Report ORNL-434, Oak Ridge National Laboratory |series=Aecd ; 2793 |year=1949 |hdl=2027/mdp.39015086443671 }}</ref><ref>{{cite journal |last1=Kahn |first1=H. |authorlink=Herman Kahn |last2=Harris |first2=T. E. |authorlink2=Theodore E. Harris |year=1949 |title=Estimation of Particle Transmission by Random Sampling |journal=Monte Carlo Method |volume=12 |series=Applied Mathematics Series |pages=27–30 |publisher=National Bureau of Standards. }}</ref> Importance sampling is also related to [[umbrella sampling]] in [[computational physics]]. Depending on the application, the term may refer to the process of sampling from this alternative distribution, the process of inference, or both.

== Basic theory ==

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== Application to probabilistic inference ==

Such methods are frequently used to estimate posterior densities or expectations in state and/or parameter estimation problems in probabilistic models that are too hard to treat analytically. Examples include [[Bayesian network]]s and importance weighted [[Variationalvariational autoencoder|variational autoencoders]]s.<ref>{{Cite journal |last1=Burda |first1=Yuri |last2=Grosse |first2=Roger |last3=Salakhutdinov |first3=Ruslan |title=Importance Weighted Autoencoders |journal=Proceedings of the 4th International Conference on Learning Representations (ICLR) |arxiv=1509.00519 |publication-date=2016}}</ref>

== Application to simulation ==