GPFlow: Gaussian Process Uncertainty Quantification I trained some Gaussian process model with the Python library GPFlow on a dataset consisting of $(X, Y)$, inputs and outputs, in a regression setting. This model gives me pretty good predictions in the sense that the relative error is small almost everywhere. I want to use the uncertainty as well, which is given in a GPFlow setting in the form of a standard deviation (STD) associated with every prediction. Here's my problem: I normalised both inputs and outputs before training (separately) using sklearn's StandardScaler (effectively making the data normally distributed with $0$ mean and unit STD). So the STD given by the model pertains to the scaled data. How do I "rescale" the uncertainty estimates of the GP to the actual data? Using the inverse_transform function of the output scaler makes little sense. This issue might be easier solvable if I scaled with a MinMaxScaler (squishing all data points into the unit interval) by dividing by the length of the range of the original output set (at least I think it works that way). But how about the case of the StandardScaler? Any insights will be appreciated!
Not sure about which function to call in GPFlow, but the principle is simple. If your $Y$ data was scaled:
$$Y_{scaled} = {(Y - mean(Y)) \over {sd(Y)}}$$
So now if you have any prediction (from the scaled model) $\hat{Y}_{scaled}$, with stand error $\sigma_{\hat {Y}_{scaled}}$, then it is a normal distribution ${\cal N}(\hat {Y}_{scaled}, \sigma^2_{\hat {Y}_{scaled}})$. Then you can very easily rescale it back to the original scale ${\cal N}(\hat {Y}, \sigma^2_{\hat {Y}})$, where:
$$\hat {Y} = \hat {Y}_{scaled}\cdot sd(Y) + mean(Y)$$
$$\sigma_{\hat {Y}} = \sigma_{\hat {Y}_{scaled}} \cdot sd(Y)$$