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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.