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Imagine that I have a set of $N$ (training) datapoints $\left\{(x_n,y_n)\right\}_{n=1}^N$, with error bars/uncertainties on each datapoint along both the $x$- and $y$-directions, written as $\left\{(\sigma_{x,n},\sigma_{y,n})\right\}_{n=1}^N$, that may not be the same between datapoints. See the image below from Hogg et al for an example of such a dataset (ignore the dotted lines).

enter image description here

My question is: is there anything in the Gaussian Process literature that deals with fitting a GP to such a 2-D uncertainty dataset? I'm confused about where the addition of the $x$ uncertainties would factor in; the covariance function? Should we also model additive noise in the $x$ direction as well, if the scatter along $x$ cannot solely be attributed to the error bars?

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  • $\begingroup$ Do you have a particular problem in mind? Can you say something about the uncertainty in $x_i$ in a probabilistic way? e.g. $x_i \sim U(l_i, u_i)$ or $x \sim N(m_i, v_i)$? $\endgroup$
    – jcken
    Jul 27, 2020 at 11:03

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