Preliminary list of abstracts

Note: We are using the fantastic MathJax JavaScript library to typeset the mathematics on this web page. You can right-click on a formula to zoom in or to select the rendering engine you like (all under 'Settings'). Under Firefox the default renderer turns out to be MathML (which is quick), but you might prefer the HTML-CSS renderer for more faithful LaTeX rendering. If you encounter any problems with this setup: then please email us!

Error estimation for the integration with respect to log-concave densities
Keywords: Markov chain Monte Carlo; Metropolis algorithm; log-concave
Wed, 09:50--10:15
• Rudolf, Daniel (Mathematical Institute, University Jena, Germany)

Let the function $f$ be given and $\rho>0$ be an unnormalized, log-concave density. Suppose that $D\subset \mathbb{R}^d$ is a convex body with additional properties. The goal is to compute the integral of $f$ with respect to the distribution determined by $\rho$, such that $S(f,\rho)=\frac{\int_D f(y) \rho(y) {\rm d}y}{\int_D \rho(x) {\rm d}x},$ is the desired quantity. A straight generation of the desired distribution, say $\mu_\rho$, is in general not possible. Thus, it is reasonable to compute the average over a suitable Markov chain $(X_n)_{n\in \mathbb{N}}$ which approximates $\mu_\rho$. Hence \begin{equation*} S_{n,n_0}(f,\rho):=\frac{1}{n} \sum_{i=1}^n f(X_{i+n_0}) \end{equation*} is the suggested approximation of $S(f,\rho)$, where $n_0$ is called burn-in. The considered Markov chains, e.g. a suitable Metropolis algorithm or Hit-and-Run algorithm, fulfill some convergence conditions. For example the Metropolis algorithm based on a ball walk has an absolute $L_2$-spectral gap, see [1]. We use these convergence properties to apply error bounds of Markov chain Monte Carlo methods to obtain an error estimate for the computation of $S(f,\rho)$, see [2,3].

[1] P. Mathé and E. Novak. Simple Monte Carlo and the Metropolis algorithm. Journal of Complexity, 23(4-6):673–696, 2007.
[2] D. Rudolf. Explicit error bounds for lazy reversible Markov chain Monte Carlo. J. Complexity, 25(1):11–24, 2009.
[3] D. Rudolf. Explicit error bounds for Markov chain Monte Carlo. PhD thesis, University Jena, 2010.