2025/04/11 by Felix Benning, Benning, Felix
Computer Science · #60A10 #60G05 #60G15 #60G60 #Advanced Decision-Making Techniques #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2504.08513
openalex publication_date 2025/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In sequential design strategies, common in geostatistics and Bayesian optimization, the selection of a new observation point Xn+1 of a random function \mathbf f is informed by past data, captured by the filtration \mathcal Fn=σ(\mathbf f(X0),…,\mathbf f(Xn)). The random nature of Xn+1 introduces measure-theoretic subtleties in deriving the conditional distribution \mathbb P(\mathbf f(Xn+1)∈ A | \mathcal Fn). Practitioners often resort to a heuristic: treating X0,…, Xn+1 as fixed parameters within the conditional probability calculation. This paper investigates the mathematical validity of this widespread practice. We construct a counterexample to prove that this approach is, in general, incorrect. We also establish our central positive result: for continuous Gaussian random functions and their canonical conditional distribution, the heuristic is sound. This provides a rigorous justification for a foundational technique in Bayesian optimization and spatial statistics. We further extend our analysis to include settings with noisy evaluations and to cases where Xn+1 is not adapted to \mathcal Fn but is conditionally independent of \mathbf f given the filtration.