# T:A:L:K:S

 close this window
 title: Error bounds for the computation of integrals by Markov chain Monte Carlo name: Rudolf first name: Daniel location/conference: SPP-JT10 PRESENTATION-link: http://www.dfg-spp1324.de/nuhagtools/event/dateien/SPP-JT10/talks/rudolf_eisenach.pdf abstract: Let the function $f$ be given and $\pi$ be a probability measure. The goal is to compute the integral of $f$ with respect to $\pi$, such that $S(f)=\int_D f(y) \pi(dy)$ is the desired quantity. A straight generation of $\pi$ is in general not possible. Thus, it is reasonable to compute the average over a Markov chain $(X_n)_{n\in \N}$ after a burn-in $n_0$. Hence \begin{equation*} \label{sum} S_{n,n_0}(f):=\frac{1}{n} \sum_{i=1}^n f(X_{i+n_0}) \end{equation*} is the suggested approximation of $S(f)$. Suppose that the Markov chain fulfills some convergence conditions, for instance has an absolute $L_2$-spectral gap $1-\beta>0$. Furthermore assume that the initial distribution $\nu$ of the Markov chain has a density with respect to $\pi$. Then it holds $\text{err}(S_{n,n_0},f)^2 \leq \frac{4 \norm{f}{\infty}^2}{n(1-\beta)} \quad \text{where} \quad n_0 = \left\lceil \frac{\log(32 \norm{\frac{d \nu}{ d \pi}-1}{2})}{1-\beta} \right\rceil.$ The bound is generalized to integrands which belong to $L_p$ where $p\in[2,\infty]$.