Can PI be used for minimization?

Yes, but the inequality and standardized difference must be reversed appropriately.

What happens when predictive standard deviation is zero?

The Gaussian division formula is not used directly. Evaluate the deterministic threshold event with an explicit equality convention.

Does a PI of 0.8 guarantee success in eight of ten evaluations?

No. It is a model-based probability for the specified event, and its reliability depends on the surrogate and evaluation setting.

AI Glossary / Probabilistic machine learning / Intermediate

Probability of improvement

Definition

Probability of improvement is an acquisition criterion used in Bayesian optimization. It scores a candidate evaluation by the model's probability that the objective there will exceed a chosen improvement threshold. The threshold is often based on the current best value plus a specified margin for a maximization problem.

What it means

When evaluating a setting is expensive, an optimizer must decide where to test next. A probabilistic surrogate supplies a distribution for the unknown objective at each candidate. Probability of improvement, often abbreviated PI, asks how likely that distribution is to beat the target.
This criterion counts success at crossing the threshold, not the size of the gain after crossing it. A candidate offering a reliable tiny improvement can therefore outrank another with a lower success probability but a much larger possible gain. The distinction matters when selecting an acquisition function for the actual search objective.

How it works

For maximization, let b be the incumbent value and xi a nonnegative improvement margin. Define PI(x) = Pr[f(x) > b + xi | data]. If the surrogate's marginal for f(x) is Gaussian with mean mu(x) and positive standard deviation sigma(x), PI(x) = Phi[(mu(x) - b - xi)/sigma(x)], where Phi is the standard normal CDF.
For minimization, reverse the direction: use Pr[f(x) < b - xi | data], giving Phi[(b - xi - mu(x))/sigma(x)] in the Gaussian case. Keep the direction consistent throughout the implementation.
Choose a candidate by optimizing this inexpensive acquisition score, evaluate the actual objective, update the surrogate, and repeat. The uncertainty must match the intended quantity: improving an underlying function is not the same event as obtaining an unusually favorable noisy observation. The incumbent also needs a defensible definition when observations are noisy.

Three practical examples

Hyperparameter search: prioritize a model configuration by its predicted chance of beating a specified validation-performance target.
Simulation tuning: select a design setting for an expensive simulator when improvement beyond a practical margin is the immediate criterion.
Experimental planning: compare candidate settings using an explicit target, while checking whether evaluation cost or constraints require a richer acquisition rule.

Think of it this way

Think of choosing which contestant is most likely to clear a bar. PI focuses on clearing the bar, regardless of whether the contestant clears it by a millimeter or a meter. This analogy is limited: the probabilities come from a surrogate that can be wrong, and choosing the best next experiment is not always equivalent to choosing the most likely immediate winner.

Worked example

Consider an original maximization example with incumbent b = 10 and improvement margin xi = 0.5, so the target is 10.5. Candidate A has a Gaussian predicted objective with mean 11 and standard deviation 0.5. Candidate B has mean 12 and standard deviation 3.
For A, the standardized gap is (11 - 10.5)/0.5 = 1, giving PI approximately 0.8413. For B, it is (12 - 10.5)/3 = 0.5, giving PI approximately 0.6915. PI chooses A despite B's higher predicted mean, because A has a greater modeled chance of clearing this particular target.
Now raise the required margin to xi = 2, making the target 12. A's standardized gap becomes -2 and PI approximately 0.0228. B's gap is zero and PI is 0.5. The ranking reverses.
These are hypothetical surrogate distributions, not empirical success rates. The example shows how the target changes the event being optimized. It does not establish which candidate has greater expected improvement or which search policy will find the best overall setting.

Illustrative case study

Fictional case: Solstice Simulation has a limited budget for testing operating settings in a simulator. The team initially uses PI with a very small margin and notices that proposed settings cluster near the current incumbent.
It reviews whether likely small improvements match the project's objective, compares the behavior with a larger margin and an expected-improvement criterion, and records the search budget for each comparison. No choice is labeled universally superior; the team assesses final results across repeated simulated searches. This invented scenario illustrates acquisition design without claiming real performance gains or guaranteed global optimization.

Common misconceptions and mistakes

PI is not expected improvement. It measures a probability, whereas expected improvement also weights the magnitude of a positive gain. It is not the probability that a candidate is globally optimal either: the event compares against one threshold, not all alternatives.
More uncertainty does not always increase PI. Its effect depends on where the predicted mean sits relative to the threshold. A larger margin can change exploration behavior, but it is not a universal cure for a poor surrogate or an unsuitable acquisition objective.

Business use

The business value of PI is its interpretable target-based selection rule. A team can state what counts as improvement and inspect why a proposed experiment is likely to meet it under the model.
Before using the score, define whether the target concerns latent performance, an observed noisy result, or a practically meaningful margin. Include experiment costs and feasibility requirements separately when they matter. Evaluate the entire optimization process by the quality of settings found within budget, rather than by how high its acquisition probabilities appear.

Check your understanding

Two candidates have the same PI but different potential improvement sizes. Does PI distinguish the larger possible gain?

Answer: No. The probability alone counts whether the target is exceeded. Another criterion is needed if improvement magnitude should affect the score.

Related terms

Bayesian optimization
Expected improvement
Gaussian process
Surrogate model
acquisition function
incumbent
exploration and exploitation
noisy objective

Learn next

Translate a maximization PI calculation into its minimization counterpart. Vary the threshold while holding candidate predictions fixed, then compare the resulting ranking with expected improvement. Finally, examine how observation noise changes the meaning of the incumbent and the event being predicted.

Expected improvement

Sources

Brochu, Cora and de Freitas, A Tutorial on Bayesian Optimization of Expensive Cost Functions, section 2.3.1 on improvement-based acquisitions: https://arxiv.org/pdf/1012.2599 Tom Rainforth, Oxford advanced machine-learning lecture on Bayesian optimization and acquisition functions: https://www.robots.ox.ac.uk/~twgr/assets/teaching/lecture_16.pdf Brochu-Cora-de Freitas Section 2.3.1 Equation 2 and noisy-incumbent discussion were independently consulted.

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