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I am having quite difficult time to clearly understand the differences between Kriging and Gaussian Process Regression. Here is what I have understood so far:

  • For simple kriging (mean value known), the two methods give the same result expect it is not from the same point of view. Simple kriging uses the best linear unbiased estimator. GPR uses the Bayesian approach by assuming a prior distribution over functions. This prior is a Gaussian process. We use a Gaussian likelihood with the observed values and that way we get the posterior distribution. We compute the expected value of this distribution and we obtain the same result than simple kriging. My first question is: How come do we get the same result as we do not make any assumption on the randomness for the best linear unbiased estimator?

  • For ordinary kriging, one does not know the mean value expect that the value is stationary. So far, I have not seen any link between the point of view using the best linear unbiased estimator and the one using Gaussian Process. Is there any?

Akusa
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  • You may be interested in https://web.stanford.edu/~hastie/Papers/ESLII.pdf section 5.8.2 Examples of RKHS, specifically this: "The estimate (5.57) also arises as the kriging estimate of a Gaussian random field in spatial statistics (Cressie, 1993). Compare also (5.58) with the smoothing spline fit (5.17) on page 154." – Adrian Aug 01 '20 at 19:47
  • Thank you very much. But in many papers, people introduce the model of kriging as $Y(x) = \mu(x) + E(x)$ where $\mu$ is deterministic and equal to the mean and $E$ is a mean-square stationary random process with zero mean and constant variance. They never said it is a Gaussian process. Is it omitted because it is obvious or not important? What I want to say is that it may be not important because in fact we do not need to exactly know this process and it is after calculus that we find that the best linear unbiased estimation correspond to this process to be Gaussian. – Akusa Aug 01 '20 at 21:24
  • For my above comment, we are in the simple kriging framework, i.e. $\mu(x)$ is constant and known. – Akusa Aug 01 '20 at 21:34

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