Simple Monte Carlo and the Metropolis algorithm
- Mathé, Peter
- Novak, Erich
2010 Mathematics Subject Classification
- 65C05 65Y20 68Q17 82B80
- Monte Carlo methods, Metropolis algorithm, log-concave density, rapidly mixing Markov chains, optimal algorithms, adaptivity, complexity
We study the integration of functions with respect to an unknown density. Information is available as oracle calls to the integrand and to the non-normalized density function. We are interested in analyzing the integration error of optimal algorithms (or the complexity of the problem) with emphasis on the variability of the weight function. For a corresponding large class of problem instances we show that the complexity grows linearly in the variability, and the simple Monte Carlo method provides an almost optimal algorithm. Under additional geometric restrictions (mainly log-concavity) for the density functions, we establish that a suitable adaptive local Metropolis algorithm is almost optimal and outperforms any non-adaptive algorithm.
- J. Complexity, 23 (2007) pp. 673--696.