Smoothing problem (stochastic processes): Difference between revisions

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The '''smoothing problem''' (not to be confused with [[smoothing]] in [[statistics]], [[image processing]] and other contexts) is the problem of [[density estimation|estimating]] an unknown [[probability density function]] recursively over time using incremental incoming measurements. It is one of the main problems defined by [[Norbert Wiener]].<ref name="wiener-report">1942, ''Extrapolation, Interpolation and Smoothing of Stationary Time Series''. A war-time classified report nicknamed "the yellow peril" because of the color of the cover and the difficulty of the subject. Published postwar 1949 [[MIT Press]]. http://www.isss.org/lumwiener.htm</ref><ref name="wiener-book">Wiener, Norbert (1949). Extrapolation, Interpolation, and Smoothing of Stationary Time Series. New York: Wiley. {{ISBN|0-262-73005-7}}.</ref> A '''smoother''' is an algorithm that implements a solution to this problem, typically based on [[recursive Bayesian estimation]]. The smoothing problem is closely related to the [[filtering problem]], both of which are studied in Bayesian smoothing theory.

A smoother is often a two-pass process, comprisedcomposed of forward and backward passes. Consider doing estimation (prediction/retrodiction) about an ongoing process (e.g. tracking a missile) based on incoming observations. When new observations arrive, estimations about past needs to be updated to have a smoother (more accurate) estimation of the whole estimated path until now (taking into account the newer observations). Without a backward pass (for [[retrodiction]]), the sequence of predictions in an online filtering algorithm does not look smooth. In other words, retrospectively, it is as if we are using future observations for improving estimation of a point in past, when those observations about future points become available. Note that time of estimation (which determines which observations are available) can be different to the time of the point that the prediction is about (that is subject to prediction/retrodiction). The observations about later times can be used to update and improved the estimations about earlier times. Doing so leads to smoother-looking estimations (retrodiction) about the whole path.

==Examples of smoothers ==