Predictive standard deviation measures how uncertain a Gaussian-process prediction is around its posterior predictive mean. Near observed training points, it can approach zero; farther away, it usually increases smoothly. In two dimensions, a heat map of this quantity reveals where the model knows less about the function. section 2.4.5
For a predictive normal distribution with mean μ and standard deviation σ, the interval μ ± βσ defines a selectable uncertainty band. β = 1, 2, and 3 correspond to approximately 68%, 95%, and 99.7% intervals, respectively. The standard deviation itself is not exactly a 95% confidence interval, but both quantify predictive uncertainty. section 5.2.2 section 2.4.5
Use predictive standard deviation to:
- Map uncertainty across a multidimensional search space alongside the predictive mean. section 2.4.5
- Build an upper-confidence-bound score μ + βσ for Bayesian optimization. section 5.2.2
- Control exploration versus exploitation: smaller β favors high predicted means, while larger β favors uncertain regions. section 5.2.2
Visualizing uncertainty over a two-dimensional grid
Train a Gaussian process on points such as (0, 0), (1, 2), and (−1, 1), evaluate a 101-by-101 grid, and reshape the resulting standard deviations into a square heat map. The map should show low uncertainty around the training observations and increasing uncertainty away from them. section 2.4.5
predictive_stddev = predictive_distribution.stddev
Extract the predictive standard deviation directly from a predictive distribution with its stddev property. section 2.4.5
