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I have a Gaussian model with mean zero, variance is arbitrary constant, and correlation function $e^{-\theta(x-x')^2}$ where $\theta$ is again an arbitrary constant.

I've plotted some realizations of the above at various different locations, and now I am supposed to use observations at alternate locations to predict the others and then compute the RMSE.

I think I have to use the conditional distribution somehow:

$$\pmatrix {{Y}\\{Y^n}}\sim N \pmatrix {{\pmatrix {{\bf f^T}\\{\bf F}}\beta,\sigma^2\pmatrix {{1}&{\bf r^T}\\{\bf r}&\bf R}}} $$

But I really don't know how I can apply my data to this, or how I would make predictions from it.

Any help would be greatly appreciated.

Edit: I've just read something about MCMC sampling. I think maybe that might be helpful?? I'm having trouble understanding it though.

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You don't need to use MCMC. You simply apply the GP prediction formula (as given in a previous question). This only involves some linear algebra.

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