5. Optimality criteria for probabilistic numerical methods
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C. J. Oates
Abstract
It is well understood that Bayesian decision theory and average case analysis are essentially identical. However, if one is interested in performing uncertainty quantification for a numerical task, it can be argued that standard approaches from the decision-theoretic framework are neither appropriate nor sufficient. Instead, we consider a particular optimality criterion from Bayesian experimental design and study its implied optimal information in the numerical context. This information is demonstrated to differ, in general, from the information that would be used in an averagecase- optimal numerical method. The explicit connection to Bayesian experimental design suggests several distinct regimes, in which optimal probabilistic numerical methods can be developed.
Abstract
It is well understood that Bayesian decision theory and average case analysis are essentially identical. However, if one is interested in performing uncertainty quantification for a numerical task, it can be argued that standard approaches from the decision-theoretic framework are neither appropriate nor sufficient. Instead, we consider a particular optimality criterion from Bayesian experimental design and study its implied optimal information in the numerical context. This information is demonstrated to differ, in general, from the information that would be used in an averagecase- optimal numerical method. The explicit connection to Bayesian experimental design suggests several distinct regimes, in which optimal probabilistic numerical methods can be developed.
Chapters in this book
- Frontmatter I
- Preface: Multivariate algorithms and information-based complexity V
- Contents IX
- 1. The control variate integration algorithm for multivariate functions defined at scattered data points 1
- 2. An adaptive random bit multilevel algorithm for SDEs 15
- 3. RBF-based penalized least-squares approximation of noisy scattered data on the sphere 33
- 4. On the power of random information 43
- 5. Optimality criteria for probabilistic numerical methods 65
- 6. ε-Superposition and truncation dimensions, and multivariate method for∞-variate linear problems 89
- 7. Adaptive approximation for multivariate linear problems with inputs lying in a cone 109
Chapters in this book
- Frontmatter I
- Preface: Multivariate algorithms and information-based complexity V
- Contents IX
- 1. The control variate integration algorithm for multivariate functions defined at scattered data points 1
- 2. An adaptive random bit multilevel algorithm for SDEs 15
- 3. RBF-based penalized least-squares approximation of noisy scattered data on the sphere 33
- 4. On the power of random information 43
- 5. Optimality criteria for probabilistic numerical methods 65
- 6. ε-Superposition and truncation dimensions, and multivariate method for∞-variate linear problems 89
- 7. Adaptive approximation for multivariate linear problems with inputs lying in a cone 109
