Probability of improvement acquisition

⚠️ Curated Limitations & Negative Results

• For a minimization incumbent b and margin xi >= 0, setting T = b - xi gives gain > xi for a strict event and gain >= xi for an inclusive event. Positive-variance Gaussian continuity makes their probabilities equal, but this equality does not extend to arbitrary predictive atoms. The zero-variance event must be evaluated directly; the printed formulas do not specify every degenerate convention.
• Snoek Section 2.2, PDF p.3 observes noisy y_n but expresses the incumbent through latent f(x_n). The inspected passage does not prescribe a unique operational noisy-incumbent estimator. A fixed data-derived threshold, an uncertain latent incumbent and a future noisy measurement define different events. A random incumbent requires its joint dependence with the prospective value; substituting its mean is not generally equivalent. No particular noisy, batch, constrained or multiobjective extension is specified here.
• The target policy is intentionally not reconstructed from the examples. Jones PDF p.21 / printed p.365 uses an observed-best target, whereas the captions on PDF pp.22–23 / printed pp.366–367 describe percentages below the surface minimum; Enhanced Method 4 separately uses a surface-minimum and observed-range formula on PDF p.23. The p.21 wording about T being too high and global search also conflicts with the decreasing-target direction on p.23. These distinctions are retained as source ambiguities, not silently unified into one schedule or presented as formal errata.
• A high PI value is conditional on the chosen predictive model and event; it is not a calibrated guarantee about the unknown objective or the remaining optimization error. Increasing predictive spread does not universally increase PI: for a fixed minimization threshold its direction depends on the mean relative to that threshold. The retained bounded examples separate probability and gain rankings without establishing a universal advantage for either acquisition.
• Hyperparameter integration in Snoek PDF p.4 concerns averaging conditional acquisition values. For a fixed event, probability averaging is valid; replacing the predictive mixture by a Gaussian using its first two moments generally changes the score. If the threshold varies with hyperparameters or represents a random incumbent, the joint event must be specified rather than identified with the same fixed-threshold probability.
• Jones PDF pp.25–26 / printed pp.369–370 derives a matching lower-confidence-bound coefficient from an attained ratio optimum with positive standard errors. That coefficient depends on the target and current predictive functions and need not be nonnegative. A fixed target is therefore not a fixed confidence-bound policy. Snoek PDF p.3 also gives an unnegated lower-confidence expression alongside a generic acquisition-maximization convention; that ancillary direction convention does not change PI Eq.1. Restricted ranking correspondences do not make PI identical to confidence-bound or entropy-information acquisitions.
• Evidence is the typeset 2001 Jones article on a university mirror and the official nine-page 2012 Snoek proceedings PDF. Publication, PDF build and acquisition metadata are distinct. The cited Kushner (1964) original PDF was not obtained after a publisher HTTP403 response, and no historical-priority or original-algorithm reconstruction claim is made. Cited convergence results and complete empirical procedures were not reproduced or adopted.
• The separate original curator calculations use exact Fraction arithmetic for finite events and ratio algebra, and floating Gaussian and entropy functions for bounded examples. They are not author-code replication, a fitted Gaussian-process optimizer, a benchmark, a general convergence proof or validation of extreme-tail numerical accuracy.