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Bayesian Optimization is a probabilistic approach to mathematical optimization using Bayes' rule to account for information gathered about the objective function.

The main advantages of such an approach is that it can account for noise in the evaluations of the objective function, and that all performed evaluations are used to constrain the optimum, not only the most recent ones.

Such an approach can be particularly useful when the objective function is (numerically or otherwise) expensive to evaluate

A traditional optimization problem can be stated as follows: given a set $A$ and a real-valued objective function $f : A \toℝ$ , we want to find an $x 0 \in A$ so that $f (x 0) \leq f (x)$ for all $x \in A$ .

In order to state the problem in a probabilistic framework, the following additional elements are needed:

a prior probability distribution $p (f)$ over the space $F$ of all possible objective functions.
a conditional probability distribution $p (y | f, x)$ predicting the probability of the outcome $y$ of an imprecise evaluation of $f$ at $x$