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    SUMMARY

  • Utility Functions
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    Probability Of Improvement

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    + gpjax.decision_making.utility_functions.probability_of_improvement + + +

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    +

    + Bases: AbstractSinglePointUtilityFunctionBuilder

    + + + + + +

    An acquisition function which returns the probability of improvement +of the objective function over the best observed value.

    +

    More precisely, given a predictive posterior distribution of the objective +function ff, the probability of improvement at a test point xx is defined as: +$PI(x)=Prob[f(x)<f(xbest)]\text{PI}(x) = \text{Prob}[f(x) < f(x_{\text{best}})]$ +where xbestx_{\text{best}} is the minimiser of the posterior mean +at previously observed values (to handle noisy observations).

    +

    The probability of improvement can be easily computed using the +cumulative distribution function of the standard normal distribution Ξ¦\Phi: +$PI(x)=Ξ¦(f(xbest)βˆ’ΞΌΟƒ)\text{PI}(x) = \Phi\left(\frac{f(x_{\text{best}}) - \mu}{\sigma}\right)$ +where ΞΌ\mu and Οƒ\sigma are the mean and standard deviation of the +predictive distribution of the objective function at xx.

    +
    References
    +

    [1] Kushner, H. J. (1964). +A new method of locating the maximum point of an arbitrary multipeak curve in the presence of noise. +Journal of Basic Engineering, 86(1), 97-106.

    +

    [2] Shahriari, B., Swersky, K., Wang, Z., Adams, R. P., & de Freitas, N. (2016). +Taking the human out of the loop: A review of Bayesian optimization. +Proceedings of the IEEE, 104(1), 148-175. doi: 10.1109/JPROC.2015.2494218

    + + + + +
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    + check_objective_present(posteriors: Mapping[str, AbstractPosterior], datasets: Mapping[str, Dataset]) -> None + +
    + + +
    + +

    Check that the objective posterior and dataset are present in the posteriors and +datasets.

    + + + +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    posteriors + Mapping[str, AbstractPosterior] + +
    +

    Dictionary of posteriors to be

    +
    +     		+ required +
    datasets + Mapping[str, Dataset] + +
    +

    Dictionary of datasets which may be used

    +
    +     		+ required +
    + + + +

    Raises:

    + + + + + + + + + + + + + +
    TypeDescription
    + ValueError + +
    +

    If the objective posterior or dataset are not present in the

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    + __init__() -> None + +
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    + build_utility_function(posteriors: Mapping[str, ConjugatePosterior], datasets: Mapping[str, Dataset], key: KeyArray) -> SinglePointUtilityFunction + +
    + + +
    + +

    Constructs the probability of improvement utility function +using the predictive posterior of the objective function.

    + + + +

    Parameters:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameTypeDescriptionDefault
    posteriors + Mapping[str, AbstractPosterior] + +
    +

    Dictionary of posteriors to be

    +
    +     		+ required +
    datasets + Mapping[str, Dataset] + +
    +

    Dictionary of datasets which may be used

    +
    +     		+ required +
    key + KeyArray + +
    +

    JAX PRNG key used for random number generation. Since

    +
    +     		+ required +
    + + + +

    Returns:

    + + + + + + + + + + + + + +
    Name TypeDescription
    SinglePointUtilityFunction + SinglePointUtilityFunction + +
    +

    the probability of improvement utility function.

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