Mini-Minimax Uncertainty Quantification for Emulators

摘要: Consider approximating a "black box" function $f$ by an emulator $\hat{f}$ based on $n$ noiseless observations of $f$. Let $w$ be point in the domain How big might error $|\hat{f}(w) - f(w)|$ be? If could arbitrarily rough, this large: we need some constraint besides data. Suppose is Lipschitz with known constant. We find lower bound number required to ensure that for best data, f(w)| \le \epsilon$. But general, will not know whether Lipschitz, much less its Assume optimistically Lipschitz-continuous smallest constant consistent maximum (over such regular $f$) possible $\hat{f}$; call "mini-minimax uncertainty" at $w$. In reality, or---if it is---it attain Hence, mini-minimax uncertainty smaller than f(w)|$. if large, then---even satisfies optimistic regularity assumption---$|\hat{f}(w) no matter how cleverly choose $\hat{f}$. For Community Atmosphere Model, $w$) set 1154~observations would single observation centroid 21-dimensional parameter space. also confidence bounds quantiles and mean over these are appreciable fraction maximum.