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## Error bounds for computing the expectation by Markov chain Monte Carlo (2009)

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Citations: | 117 - 2 self |

### Citations

718 | Monte Carlo Strategies in scientific computing - Liu - 2001 |

602 |
Interpolation of operators
- Bennett, Sharpley
- 1988
(Show Context)
Citation Context ...∞. The norm of P as operator on ℓ0 2 and ℓ04 is essential in the analysis. We state and show some results which are implied by the Theorem of Riesz-Thorin. For a proof and an introduction we refer to =-=[BS88]-=-. Proposition 1 (Theorem of Riesz-Thorin). Let 1 ≤ p, q1, q2 ≤ ∞. Further let θ ∈ (0, 1) and and Then 1 1 − θ := + p q1 θ q2 T : ℓq1 → ℓq1 with ‖T ‖ ℓq1 →ℓq 1 T : ℓq2 → ℓq2 with ‖T ‖ ℓq2 →ℓq 2 ‖T ‖ ℓp... |

379 | Markov chains and mixing times - Levin, Peres, et al. - 2009 |

348 | Geometric Bounds for Eigenvalues of Markov Chains
- Diaconis, Stroock
- 1991
(Show Context)
Citation Context ...e of the Markov chain to stationarity. In general it is not easy to handle β1 or β, but there are different auxiliary tools, e.g.6 DANIEL RUDOLF canonical path technique, conductance (see [JS89] and =-=[DS91]-=-), logSobolev inequalities and path coupling. For a small survey see [Ran06]. 2.3. Norm of the transition matrix. Let us consider P and S as operators acting on ℓp. Then the functional S maps arbitrar... |

345 | Approximating the Permanent
- Jerrum, Sinclair
- 1989
(Show Context)
Citation Context ... convergence of the Markov chain to stationarity. In general it is not easy to handle β1 or β, but there are different auxiliary tools, e.g.6 DANIEL RUDOLF canonical path technique, conductance (see =-=[JS89]-=- and [DS91]), logSobolev inequalities and path coupling. For a small survey see [Ran06]. 2.3. Norm of the transition matrix. Let us consider P and S as operators acting on lp. Then the functional S ma... |

178 | Hierarchical Singular Value Decomposition of Tensors - Grasedyck - 2009 |

177 | General state space Markov chains and MCMC algorithms
- Roberts, Rosenthal
(Show Context)
Citation Context ...nce of the bounds of (ii) and (iii) in Theorem 11 on the initial distribution is encouraging for an extension to general state spaces. (For an introduction to MCMC on general state spaces we refer to =-=[RR04]-=-.) But the dependence of the initial distribution on the estimate in the l2-case is disillusioning because of the additional factor of ∥ 1∥ . ∞ π In [Rud09, Theorem 8, p.10] a similar l∞-bound of Sn,n... |

175 | Monte Carlo methods in statistical mechanics: Foundations and new algorithms, lectures given at the Cours de Troisiéme Cycle de la Physique en Suisse Romande - Sokal - 1989 |

138 | Finite Markov chains and algorithmic applications, volume 52 - Häggström - 2002 |

137 | The Easy Path Wavelet Transform: A New Adaptive Wavelet Transform for Sparse Representation of Two-dimensional Data - Plonka - 2008 |

137 | Dual Pricing of Multi-Exercise Options under Volume Constraints - Bender - 2009 |

134 | Optimal Approximation of Elliptic Problems by Linear and Nonlinear - Dahlke, Novak, et al. - 2009 |

133 | Approximation of Infinitely Differentiable Multivariate - Novak, Woźniakowski - 2009 |

131 | Greedy Solution of Ill-Posed Problems: Error Bounds and Exact Inversion - Denis, Lorenz, et al. - 2009 |

129 | Black Box Low Tensor Rank Approximation Using Fibre-Crosses - Espig, Grasedyck, et al. - 2008 |

128 | A Review of Curvelets and Recent Applications - Ma, Plonka - 2009 |

128 | Minimization of Non-smooth, Non-convex Functionals by Iterative Thresholding - Bredies, Lorenz - 2009 |

128 | Regularization with Non-convex Separable Constraints - Bredies, Lorenz - 2009 |

124 | A Compressive Landweber Iteration for Solving Ill-Posed Inverse Problems - Ramlau, Teschke, et al. - 2008 |

122 | Adaptive Wavelet Methods and Sparsity Reconstruction for Inverse Heat Conduction Problems - Bonesky, Dahlke, et al. - 2009 |

120 | Optimal Order of Convergence and (In-) Tractability of Multivariate Approximation of Smooth Functions - Novak, Woźniakowski - 2008 |

119 | Curvelet-Wavelet Regularized Split Bregman Iteration for Compressed Sensing - Plonka, Ma - 2009 |

118 | A Two Parameter Generalization of Lions’ Nonoverlapping Domain Decomposition Method for Linear Elliptic PDEs - Friedrich - 2009 |

116 | Accelerated Projected Steepest Descent Method for Nonlinear Inverse Problems with Sparsity Constraints - Teschke, Borries - 2009 |

115 | Variable Subspace Sampling and Multi-level Algorithms - Müller-Gronbach, Ritter - 2009 |

115 | Optimally Sparse Image Representation by the Easy Path Wavelet Transform - Plonka, Tenorth, et al. - 2009 |

113 | Nonequispaced Hyperbolic Cross Fast Fourier Transform - Döhler, Kunis, et al. - 2009 |

48 | Markov chain decomposition for convergence rate analysis. - Madras, Randall - 2002 |

33 | Rapidly mixing Markov chains with applications in computer science and physics. - Randall - 2006 |

23 | Optimal spectral structure of reversible stochastic matrices, Monte Carlo methods and the simulation of Markov random - Frigessi, Hwang, et al. - 1992 |

17 | Markov chains and mixing - Levin, Peres, et al. - 2009 |

17 | Geometric ergodicity and hybrid markov chains. Elect - Roberts, Rosenthal - 1997 |

14 | Examples comparing importance sampling and the Metropolis algorithm - Bassetti, Diaconis - 2005 |

12 | Explicit error bounds for lazy reversible Markov chain Monte Carlo
- Rudolf
(Show Context)
Citation Context ...o do a burn-in, i.e. n0 = 0. Secondly we relate the result of the first step to the general case where the chain is initialized by a distribution ν. The techniques which we will use are similar as in =-=[Rud09]-=-. 3.1. Starting from stationarity. This is also called starting in equilibrium, i.e. the distribution of the Markov chain does not change, it is already balanced. In the following we will always denot... |

6 | Markov chains, Texts in Applied Mathematics - Brémaud - 1999 |

3 | Numerical integration using Markov chains, Monte Carlo Methods - Mathé - 1999 |

2 | Monte Carlo Strategies in Scientific Computing - Jun - 2001 |

1 | Metropolis algorithm and equienergy sampling for two mean field spin systems - Bassetti, Leisen - 2007 |

1 |
Numerical integration using Markov chains, Monte Carlo Methods Appl. 5
- times, Providence
- 1999
(Show Context)
Citation Context ...β1 or β, but there are different auxiliary tools, e.g.6 DANIEL RUDOLF canonical path technique, conductance (see [JS89] and [DS91]), logSobolev inequalities and path coupling. For a small survey see =-=[Ran06]-=-. 2.3. Norm of the transition matrix. Let us consider P and S as operators acting on lp. Then the functional S maps arbitrary functions to constant functions. Let l 0 p := l 0 p(D, π) = {g ∈ lp : S(g)... |