Fast computation of expectations and integrals by Markov chain Monte Carlo methods; error bounds and burn-in

我们研究了高维域或复杂结构域上加权和和积分数值计算的快速算法。我们希望以最优的方式调整算法，这取决于被积函数的性质，密度和定义域。加权和和积分在许多应用中都有，特别是在统计物理、统计学、金融数学和计算机科学中。令人惊讶的是，Metropolis算法是最广泛使用的算法之一，但是，已知的错误界限并不多。对于应用程序，停止标准以及预热时间是最重要的。由于缺乏（非渐近）误差界，这些算法的重要部分通常是启发式选择的。相反，我们证明了显式的误差界限。现在我们将进一步研究这些边界的应用，以及改进的误差边界和算法。

We study fast algorithms for the numerical computation of weighted sums and integrals over domainsthat are high dimensional or have a complicated structure. We want to tune the algorithms in an optimal way, depending on properties of the integrand, the density, and the domain. Weighted sums and integrals are met in numerous applications, especially in statistical physics, in statistics, in financial mathematics and in computer science. It is amazing that the Metropolis algorithm is one of the most widely used algorithms but, nevertheless, not many error bounds are known. For the applications, stopping criteria as well as the time for warming up are most important. Because of the lack of (nonasymptotic) error bounds, these important parts of the algorithms are usually chosen heuristically. In contrast, we proved explicit error bounds. Now we will study further applications of these bounds as well as modified error bounds and algorithms.