Wavelet Monte Carlo: a principle for sampling from complex distributions

摘要

Abstract We present Wavelet Monte Carlo (WMC), a new method for generating independent samples from complex target distributions. The methodology is based on wavelet decomposition of the difference between the target density and a user-specified initial density, and exploits both wavelet theory and survival analysis. In practice, WMC?can process only a finite range of wavelet scales. We prove that the resulting L1documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L_1$$end{document} approximation error converges to zero geometrically as the scale range tends to (-∞,+∞)documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(-infty ,+infty )$$end{document}. This provides a principled approach to trading off accuracy against computational efficiency. We offer practical suggestions for addressing some issues of implementation, but further development is needed for a computationally efficient methodology. We illustrate the methodology in one- and two-dimensional examples, and discuss challenges and opportunities for application in higher dimensions.