The effect of ScaleKernel in PairwiseGP

[Thomasq99]

Hi everyone,

I have a question regarding the way PairwiseGP handles the likelihood by implicitly setting noise to 1 ($\sigma=1$ ). Instead of having $\sigma$ as a hyperparameter, pairwiseGP simply sets it to 1 and uses ScaleKernel(Kernel).

Does this mean that if the actual noise in stated preference relations is 0.1 that the scale kernel will scale by 10? How does this scaleKernel handle setting the noise to 1?

Additionally, in the cited paper in the code (Chu, W., & Ghahramani, Z. (2005, August). Preference learning with Gaussian processes. In Proceedings of the 22nd international conference on Machine learning (pp. 137-144).), the predictive preference (equation 19) includes the sigma value. If I want to compute the predictive preference, would I still need to use $\sigma$ or is it handled by the scale kernel?

[Balandat]

At a high level, using both a scale kernel and noise variance would mean the model would be overparameterized. @ItsMrLin should be able to provide more details on the reasoning here!

[Thomasq99]

Thank you! I managed to find out the reasoning. I could not find a way to delete the discussion.