Wavelet Monte Carlo: a principle for sampling from complex distributions

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 $$L_1$$ L 1 approximation error converges to zero geometrically as the scale range tends to $$(-\infty ,+\infty )$$ ( - ∞ , + ∞ ) . 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.