The derivation of gradient of the marginal likelihood w.r.t covariance function hyperparameters $\theta$ is given in http://www.gaussianprocess.org/gpml/chapters/RW5.pdf, page 125. However, the gradient w.r.t likelihood parameters (let's call them $\sigma$) is not derived.

I found the final equations for this gradient w.r.t $\sigma$ in the source code of the GPML toolbox by the book's authors and the GPStuff toolbox, as well as in this paper. However, I found no derivation. The equations are similar to those of the gradient w.r.t the covariance function hyperparameters $\theta$. But I don't understand how $\frac{\partial \hat{f}}{\partial \sigma}$ is derived (supposedly similar to $\frac{\partial \hat{f}}{\partial \theta}$ in Eq. (5.24) of the GPML book). The abovementioned paper (Eq. (18)) and the source code of both GPML and GPstuff all use the equation: $$\frac{\partial \hat{f}}{\partial \sigma} = (I + K W)^{-1} K \frac{\partial}{\partial \sigma} \nabla \log p(y | \hat{f}) \text.$$ I tried but could not figure out how to derive this same equation.

Any reference or hint to derive this equation is appreciated. Thanks!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.