The difference between smoothing problems in stochastic processes and statistics

Question

I am taking a class and we are about to talk about Kalman filters and smoothing and I am trying to do some reading ahead.

On the Wikipedia page for 'Smoothing (stochastic processes)' it says 'the smoothing problem (not to be confused with smoothing in statistics, image processing and other contexts) is the problem of estimating an unknown probability density function recursively over time using incremental incoming measurements.' On the page for Smoothing in the statistical context it says, 'In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise or other fine-scale structures/rapid phenomena.' I am a bit confused about the difference between these two topics. Especially because on both pages Kalman filters are listed as algorithms that are used for smoothing and filtering. Is smoothing in statistics just a special case of the more general smoothing problem in stochastic processes?

Any clarification or reading advice is greatly appreciated!

Answer 1

If you are coming from a pure mathematical background, it might be good to read the History section of the Kalman filtering page as the origins in control theory are good to put it in context. Reading the pages for state-space model, state-observer, LQR control and LQG control might be good background if they aren't familiar already.

For the difference between smoothing in stochastic processes vs. Other areas, the main issue I think they are getting at is that the data for time series should (probably) be causal, i.e. the present can't depend on the future. On the other hand in spatial problems, for instance, data is free to be related to any other data you like.