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# 1. Acquisition

Function to minimize over the posterior distribution.

## 1.1. `gaussian_lcb`

Use the lower confidence bound to estimate the acquisition values.

The trade-off between exploitation and exploration is left to be controlled by the user through the parameter `kappa`.

## 1.2. `gaussian_pi`

Use the probability of improvement to calculate the acquisition values.

The conditional probability `P(y=f(x) | x)` form a gaussian with a certain mean and standard deviation approximated by the model.

The PI condition is derived by computing `E[u(f(x))]` where `u(f(x)) = 1`, if `f(x) < y_opt` and `u(f(x)) = 0`, if `f(x) > y_opt`.

This means that the PI condition does not care about how “better” the predictions are than the previous values, since it gives an equal reward to all of them.

Note that the value returned by this function should be maximized to obtain the `X` with maximum improvement.

## 1.3. `gaussian_ei`

Use the expected improvement to calculate the acquisition values.

The conditional probability `P(y=f(x) | x)` form a gaussian with a certain mean and standard deviation approximated by the model.

The EI condition is derived by computing `E[u(f(x))]` where `u(f(x)) = 0`, if `f(x) > y_opt` and `u(f(x)) = y_opt - f(x)`, if `f(x) < y_opt`.

This solves one of the issues of the PI condition by giving a reward proportional to the amount of improvement got.

Note that the value returned by this function should be maximized to obtain the `X` with maximum improvement.