So I am learning Bayesian Optimization and came across expected improvement.
My question is are we searching for the point in the Gaussian Process model whose expected value (determined by mean and confidence) shall be decreased the most if sampled at that point? So is the starting criteria is to take the lowest point in GP and from there determine what is the next point that whose expected value is lowest then any other point in the GP?
How do we intuitively quantify expected improvement distribution $\phi$ in the graph attached?
As far as I know, in practice, the first observations in Bayesian optimization are random before the Gaussian processes take over. After the initial observations, the expected improvement can be calculated for a data point x. By doing so, we want to select the value for x that is expected to improve the results of our objective function the most.