# T:A:L:K:S

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 title: Hit-and-run for numerical integration name: Rudolf first name: Daniel location/conference: SPP-JT11 PRESENTATION-link: http://www.dfg-spp1324.de/nuhagtools/event_NEW/dateien/SPP-JT11/talks/SPP-JT2011_08_rudolf.pdf abstract: The hit-and-run algorithm provides a Markov chain which can be used to approximate distributions which might be given by unnormalized densities. Assume that $\mu_\rho$ is such a distribution given by $\rho$. Then we want to compute $S(f,\rho) =\int_D f(y)\, \mu_\rho({\rm d}y) =\frac{\int_D f(y) \rho(y)\, {\rm d}y}{\int_D \rho(y)\, {\rm d}y},$ for a function $f$ and $D\subset \mathbb{R}^d$. Under suitable assumptions on the densities one has explicit estimates of the total variation distance of the hit-and-run distribution to $\mu_\rho$. These estimates are used to get error bounds for Markov chain Monte Carlo methods which depend polynomially on the dimension $d$.