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In spatial data analysis, a simple way to model the covariance stucture between spatial observations is via a covariance function like $cov(y_i,y_j) = C e^{-rD_{ij}}$, based on some (euclidean) distance matrix $D$. Bayesian parameter estimation via MCMC would require repeatedly inverting the changing covariance matrix, which is computationally impossible for large N.

Is there a kind of standard solution to this problem? Is it possible to directly parametrize the Cholesky decomposition in term of the parameters, to circumvent the matrix inversion?

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There are ways to update Cholesky decompositions, but I don't understand enough about your problem from your scant description to understand if that would help.

It's also possible to work with (and to parameterize) a precision matrix rather than a covariance matrix, but again I don't understand enough about your problem from your scant description to understand if that would help.

If you could clarify, it might be easier to say more.

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Not really.

the curse of dimensionality, or what is called "the big n problem", is a serious problem in spatial statistics. There are ways to overcome it but working directly with the Cholesky decomposition of the covariance matrix is not one of them.

If you want to learn the various ways I suggest you to read this review http://link.springer.com/article/10.1007%2Fs10260-012-0207-2#page-1

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