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I'm trying to figure out how to calculate the cross-terms of the covariance matrix between the function values and derivatives of a Gaussian Process. For context, this is needed for Gaussian Process regression with positivity and monotonicity constraints, where the derivative and/or function values are constrained (see Da Veiga and Marrel 2020).

Specifically, this means calculating the posterior (predictive) covariance between the function values and derivative at a set of 'constraint points' $X_z$: $$\mathrm{Cov}(f(X_z), \frac{\partial f(X_z)}{\partial x}) $$

So far I have found this link (which references a pdf written by McHutchon for which the URL is now broken) for calculating the variance of the derivative of the Gaussian Process $\mathrm{Cov}(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial x})$, but I've struggled to derive any expression for the combined covariance of $f(x)$ and $\frac{\partial f}{\partial x}$

Eddy
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  • See section 9.4 of GP4ML https://gaussianprocess.org/gpml/chapters/RW9.pdf – jcken Oct 04 '22 at 08:46
  • @jcken Thanks, but that only gives the expression for the prior covariance of the function values with the derivatives, not the posterior. I have to condition the GP on given data points X_s (more specifically, calculate Cov(f, df/dx | f(X_s) = f_s) – Eddy Oct 04 '22 at 09:47
  • You can just apply the standard GP posterior variance; $var(A|B) = var(A) - cov(A, B) var(B)^{-1} cov(B, A)$. Use $A = (f, \frac{df}{dx})$, $B = f(X_s)$. Then $Cov(f, \frax{df}{dx} | f(X_s)$, the posterior covariance, would then be read off as a `corner' (upper right block)of $var(A|B)$ – jcken Oct 04 '22 at 09:55

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