# How does Gaussian process update after adding a new point to the model?

I am using a Gaussian Process model in the Bayesian Optimization setting. Concretely, a gaussian process is built on some initial $$N$$ points and the model is updated sequentially by evaluating a new point $$x$$.

I had a hypothesis to test that the predicted mean and covariance value of some unseen point doesn't change relatively 'much' across iterations. Could you please let me know if we could quantify how much is the change based on the kernel matrix $$K$$ or something else?