Calculator Inputs
Enter surrogate model outputs, current best value, and exploration bias. The form uses a responsive grid with three columns on large screens.
Example Data Table
Example below assumes a maximization task with current best value 0.82 and exploration bias 0.01.
| Candidate | Predictive Mean | Std. Deviation | Improvement Margin | Z Score | PI (%) |
|---|---|---|---|---|---|
| Model Alpha | 0.7900 | 0.0500 | -0.0400 | -0.8000 | 21.19 |
| Model Beta | 0.8400 | 0.0400 | 0.0100 | 0.2500 | 59.87 |
| Model Gamma | 0.8800 | 0.0800 | 0.0500 | 0.6250 | 73.40 |
| Model Delta | 0.8100 | 0.1200 | -0.0200 | -0.1667 | 43.38 |
Formula Used
For maximization:
PI(x) = Φ((μ(x) − f* − ξ) / σ(x))
For minimization:
PI(x) = Φ((f* − μ(x) − ξ) / σ(x))
Where:
- μ(x) is the predictive mean from the surrogate model.
- σ(x) is the predictive standard deviation.
- f* is the current best observed objective value.
- ξ controls exploration pressure.
- Φ is the standard normal cumulative distribution function.
A larger PI means a greater chance that the candidate will outperform the current best after accounting for uncertainty and exploration preference.
How to Use This Calculator
- Enter a candidate name so exported summaries remain clear.
- Provide the predictive mean from your Gaussian-process or surrogate model.
- Enter the predictive standard deviation for the same candidate.
- Supply the current best observed objective value.
- Set the exploration bias. Larger values demand stronger improvement.
- Select whether your task is maximization or minimization.
- Submit the form to view PI, risk, z score, sensitivity, and the Plotly graph.
- Use CSV or PDF export to save the result for reports or model reviews.
Frequently Asked Questions
1) What does probability of improvement measure?
It measures the chance that a candidate point will beat the current best value under a probabilistic surrogate model and chosen exploration bias.
2) When is PI useful in machine learning?
PI is useful in Bayesian optimization, hyperparameter tuning, black-box search, and experimental design where model predictions include uncertainty estimates.
3) Why does exploration bias matter?
Exploration bias raises the improvement bar. Larger values reduce PI unless the candidate mean is strong enough to justify more adventurous sampling.
4) What happens when standard deviation is zero?
The candidate becomes deterministic. PI becomes either 100% or 0%, depending on whether the predicted mean clears the required improvement threshold.
5) How is PI different from expected improvement?
PI only asks whether improvement is likely. Expected improvement also measures how large that improvement might be, not just its probability.
6) Should a higher PI always be chosen?
Not always. A point with slightly lower PI may still be better if it offers larger expected gains or broader exploration value.
7) Can this calculator handle minimization tasks?
Yes. Select minimization and the formula reverses the comparison so lower objective values are treated as improvements.
8) How should I interpret the chart?
The chart shows how PI changes as predictive mean moves around the current candidate while uncertainty, best value, and exploration bias remain fixed.