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I am reading "Gaussian Process Training with Input Noise" by Andrew McHutchon and Carl Edward Rasmussen, where it is assumed that the inputs $x$ are noisy measurements of the actual latent input $\tilde{x}$ , i.e $ x = \tilde{x}+\epsilon_x$ with $\epsilon_x\sim N(0,\Sigma_x)$. The observation is: $y=f(\tilde{x}+\epsilon_x)+\epsilon_y$ under a GP model $f$.

The authors consider taylor expansion of the above expression up to the first order terms as follows: $y = f(x) + \epsilon_x^T\delta_\bar{f}+\epsilon_y$, where $\delta_\bar{f}$ represents the derivative of the mean of the GP function. I`d like to know why this 1st order expansion would be an accurate-enough approximation for the input error propagation.

If anyone could help, I would be very grateful.

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