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I am doing a course at university and it deals with Gaussian Processes mainly. We use them for fitting data and prediction, machine learning, regression, classification. Is there any particular reason why Gaussian Processes stand out from the millions other random processes you could have?

I suppose that's a bit like asking does the Normal distribution stand out from the million other distributions you can have but...There is a website http://www.gaussianprocess.org/ for the ones interested.

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central limit theorem is one reason Normal distribution stands out. –  ShawnD Feb 18 '12 at 0:52
    
i believe that normal disto's are essentially unique in being determined by their mean and standard deviation. This has huge practical computational advantages. Finally they seem to 'occur in nature' which makes them important because they are useful for real world modeling. –  aginensky Feb 18 '12 at 1:59

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Gaussian processes are much easier to analyze, so are useful to produce a first approximation to reality. A frequent mistake (see, eg, recent derivative mispricing debacles) is to confuse first approximation for the truth.

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+1 Couldn't agree more. Sometimes, I think it is a miracle that anything works at all. If I had a dollar for everytime I read, ''Assume that the noise is additive white and Gaussian'', I would be a rich man. –  Robby McKilliam Jun 13 '12 at 0:22
    
I like your last sentence---amazingly, people keep making this convenient mistake! –  Suvrit Jun 13 '12 at 3:54

A well-known probabilist once told me that the 3 main classes of stochastic processes are Gaussian, Markov, and martingales.

Martingales are definitely useful in finance and also with respect to other betting games, but it seems to me that they are essential mostly because we have such powerful tools to study them. These tools have often proved useful in much broader contexts, for example the martingale problem/diffusions-- but in my own humble opinion, Gaussian and Markov processes are more natural.

As for Markov processes- the Markov property is clearly very natural. But again IMHO, these processes are most useful in modeling short time behavior, before central limit behavior has kicked in. Although, again stationary distributions and long-time behavior of Markov processes absent Gaussianity are definitely well worth studying (and of course the fruitful study of the mixing of Markov chains connects short term to long term behavior)

Gaussian processes are mostly used to study correlations and dependence, and one might argue that the notions of independence and dependence are what really make probability different from other fields (every remark has its caveat, and the connection between correlations and positive definite functions via Bochner's thm allows analysts to get in this game). Since anything with finite variance converges in a sense to a Gaussian, the idea is that Gaussian processes allow us to study how dependence behaves in the long run (for infinite variance, one simply replaces Gaussian with stable).

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and Gaussian Markov martingales are Brownian motion, mostly. –  Will Sawin Jun 13 '12 at 2:20

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