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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2504.08513 |
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| _version_ | 1866908827812626432 |
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| author | Benning, Felix |
| author_facet | Benning, Felix |
| contents | In sequential design strategies, common in geostatistics and Bayesian optimization, the selection of a new observation point $X_{n+1}$ of a random function $\mathbf f$ is informed by past data, captured by the filtration $\mathcal F_n=σ(\mathbf f(X_0),\dots,\mathbf f(X_n))$. The random nature of $X_{n+1}$ introduces measure-theoretic subtleties in deriving the conditional distribution $\mathbb P(\mathbf f(X_{n+1})\in A \mid \mathcal F_n)$. Practitioners often resort to a heuristic: treating $X_0,\dots, X_{n+1}$ 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 $X_{n+1}$ is not adapted to $\mathcal F_n$ but is conditionally independent of $\mathbf f$ given the filtration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08513 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Measure Theory of Conditionally Independent Random Function Evaluation Benning, Felix Probability Statistics Theory 60A10, 60G05, 60G15, 60G60 In sequential design strategies, common in geostatistics and Bayesian optimization, the selection of a new observation point $X_{n+1}$ of a random function $\mathbf f$ is informed by past data, captured by the filtration $\mathcal F_n=σ(\mathbf f(X_0),\dots,\mathbf f(X_n))$. The random nature of $X_{n+1}$ introduces measure-theoretic subtleties in deriving the conditional distribution $\mathbb P(\mathbf f(X_{n+1})\in A \mid \mathcal F_n)$. Practitioners often resort to a heuristic: treating $X_0,\dots, X_{n+1}$ 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 $X_{n+1}$ is not adapted to $\mathcal F_n$ but is conditionally independent of $\mathbf f$ given the filtration. |
| title | Measure Theory of Conditionally Independent Random Function Evaluation |
| topic | Probability Statistics Theory 60A10, 60G05, 60G15, 60G60 |
| url | https://arxiv.org/abs/2504.08513 |