botorch/acquisition/analytic.py [73:113]:
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    r"""Single-outcome Expected Improvement (analytic).

    Computes classic Expected Improvement over the current best observed value,
    using the analytic formula for a Normal posterior distribution. Unlike the
    MC-based acquisition functions, this relies on the posterior at single test
    point being Gaussian (and require the posterior to implement `mean` and
    `variance` properties). Only supports the case of `q=1`. The model must be
    single-outcome.

    `EI(x) = E(max(y - best_f, 0)), y ~ f(x)`

    Example:
        >>> model = SingleTaskGP(train_X, train_Y)
        >>> EI = ExpectedImprovement(model, best_f=0.2)
        >>> ei = EI(test_X)
    """

    def __init__(
        self,
        model: Model,
        best_f: Union[float, Tensor],
        posterior_transform: Optional[PosteriorTransform] = None,
        maximize: bool = True,
        **kwargs,
    ) -> None:
        r"""Single-outcome Expected Improvement (analytic).

        Args:
            model: A fitted single-outcome model.
            best_f: Either a scalar or a `b`-dim Tensor (batch mode) representing
                the best function value observed so far (assumed noiseless).
            posterior_transform: A PosteriorTransform. If using a multi-output model,
                a PosteriorTransform that transforms the multi-output posterior into a
                single-output posterior is required.
            maximize: If True, consider the problem a maximization problem.
        """
        super().__init__(model=model, posterior_transform=posterior_transform, **kwargs)
        self.maximize = maximize
        if not torch.is_tensor(best_f):
            best_f = torch.tensor(best_f)
        self.register_buffer("best_f", best_f)
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botorch/acquisition/analytic.py [203:241]:
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    r"""Single-outcome Probability of Improvement.

    Probability of improvement over the current best observed value, computed
    using the analytic formula under a Normal posterior distribution. Only
    supports the case of q=1. Requires the posterior to be Gaussian. The model
    must be single-outcome.

    `PI(x) = P(y >= best_f), y ~ f(x)`

    Example:
        >>> model = SingleTaskGP(train_X, train_Y)
        >>> PI = ProbabilityOfImprovement(model, best_f=0.2)
        >>> pi = PI(test_X)
    """

    def __init__(
        self,
        model: Model,
        best_f: Union[float, Tensor],
        posterior_transform: Optional[PosteriorTransform] = None,
        maximize: bool = True,
        **kwargs,
    ) -> None:
        r"""Single-outcome analytic Probability of Improvement.

        Args:
            model: A fitted single-outcome model.
            best_f: Either a scalar or a `b`-dim Tensor (batch mode) representing
                the best function value observed so far (assumed noiseless).
            posterior_transform: A PosteriorTransform. If using a multi-output model,
                a PosteriorTransform that transforms the multi-output posterior into a
                single-output posterior is required.
            maximize: If True, consider the problem a maximization problem.
        """
        super().__init__(model=model, posterior_transform=posterior_transform, **kwargs)
        self.maximize = maximize
        if not torch.is_tensor(best_f):
            best_f = torch.tensor(best_f)
        self.register_buffer("best_f", best_f)
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