How to compute Expected Improvement distribution from covariance- and mean-functions

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I am recreating the 2014 freeze thaw-bayesian optimization. Because of the intricate model, i cannot use standard kernels and GPs, but i have (costly) functions that manually calculate the mean and covariances for any given vector of x’s. This corresponds to having the posterior predictive distribution given as function. (Equation 20 or 21 in the paper
How do i (efficiently) compute EI on this?

(Sampling? And how so? Or a grid and how fine should that be?)

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