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So I know there are several types of acquisition function for Bayesian optimization technique. But according to Wikipedia

There are several methods used to define the prior/posterior distribution over the objective function. The most common two methods use Gaussian processes in a method called Kriging. Another less expensive method uses the Parzen-Tree estimator to construct two distributions for 'high' and 'low' points, and then finds the location that maximizes the expected improvement.

So is the expected improvement a Gaussian process, or is it something different?

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Expected improvement (EI) is a type of acquisition function. However, it relies on a surrogate model which often is defined as Gaussian processes. So, they are different things. While the surrogate model helps to represent the underlying (true) objective function of the problem, acquisition function like EI is oriented to help in selecting the next point to evaluate in the true objective function.

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