Part 2 · Intuition

Uncertainty is the resource

What does the posterior actually give you at every candidate x? Two numbers: a mean μ(x) — the model’s best guess of f(x) — and a standard deviation σ(x) — how wrong that guess could plausibly be. Near observed points, σ collapses (with noiseless observations, to exactly zero: the posterior interpolates). Far from data, σ swells back toward the prior. This “map of ignorance” is what separates BO from every model-free method: greedy on μ alone re-samples the neighborhood of the best point found and gets stuck; greedy on σ alone is uniform space-filling that ignores everything learned. Every acquisition function is some principled blend of the two.