A confidence interval summarizes uncertainty around an unknown quantity. In Gaussian-process modeling, it surrounds the mean prediction and helps decide where to evaluate an expensive black-box objective. section 1.4 section 2.4.4
For a predictive normal distribution with mean μ and standard deviation σ, a commonly used 95% interval is μ ± 1.96σ. Compute σ by taking the square root of each predictive covariance diagonal entry; the resulting lower and upper endpoints define the interval. section 2.4.4
Changing the standard-deviation multiplier changes both coverage and decision behavior:
- β = 1 gives the 68% interval, β = 2 gives approximately the 95% interval, and β = 3 gives the 99.7% interval. section 5.2.2
- In upper confidence bound optimization, μ + βσ becomes the acquisition score: smaller β favors exploitation, while larger β favors exploration. section 5.2.1 section 5.2.2
Plotting a GP confidence interval
Use a predictive distribution's mean for the center line and confidence_region() for the lower and upper bounds. Plot the bounds as a shaded region around the mean; individual samples can reveal nonlinear functions that the interval summarizes. section 2.4.4
predictive_lower, predictive_upper =
predictive_distribution.confidence_region()
Retrieve the lower and upper confidence-interval endpoints directly from a GPyTorch predictive distribution. section 2.4.4
