I am working on an active learning problem involving training a surrogate model using a GPR. I am scaling the data set using a standard scaler (it scales the data to a normal with 0 mean and standard deviation of 1). I have a train set and a candidate set and I have to choose the next suggestion using a predictive entropy function which computes a log(det|K|) where K is the covariance matrix.
Now covariance matrix is in the space of transformed random variables. I am not sure how to scale it back to the original scale. Assuming the scaled variance (scaler_y.var_), I was thinking it would be a point wise multiplication:
K_(original) = K*(scaler_y.var_**0.5 @ (scaler_y.var_.T)**0.5)
I am not sure if this is right. Any inputs on this would help? And also it would be helpful to have some suggestions for theory proving this is how the variance/covariance matrix should be scaled back.
I need help with the correct formula for computing it.