Random generation of combinatorial structures from a uniform
Theoretical Computer Science
Journal of Complexity
A Chernoff Bound for Random Walks on Expander Graphs
SIAM Journal on Computing
Rapidly Mixing Markov Chains with Applications in Computer Science and Physics
Computing in Science and Engineering
Simple Monte Carlo and the Metropolis algorithm
Journal of Complexity
Explicit error bounds for lazy reversible Markov chain Monte Carlo
Journal of Complexity
Monte Carlo Strategies in Scientific Computing
Monte Carlo Strategies in Scientific Computing
On Monte Carlo methods for Bayesian multivariate regression models with heavy-tailed errors
Journal of Multivariate Analysis
Monte Carlo Statistical Methods
Monte Carlo Statistical Methods
Optimal Monte Carlo integration with fixed relative precision
Journal of Complexity
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We assume a drift condition towards a small set and bound the mean square error of estimators obtained by taking averages along a single trajectory of a Markov chain Monte Carlo algorithm. We use these bounds to construct fixed-width nonasymptotic confidence intervals. For a possibly unbounded function f:X-R, let I(f)=@!"Xf(x)@p(dx) be the value of interest and I@?"t","n(f)=(1/n)@?"i"="t^t^+^n^-^1f(X"i) its MCMC estimate. Precisely, we derive lower bounds for the length of the trajectory n and burn-in time t which ensure that P(|I@?"t","n(f)-I(f)|@?@e)=1-@a. The bounds depend only and explicitly on drift parameters, on the V-norm of f, where V is the drift function and on precision and confidence parameters @e,@a. Next we analyze an MCMC estimator based on the median of multiple shorter runs that allows for sharper bounds for the required total simulation cost. In particular the methodology can be applied for computing posterior quantities in practically relevant models. We illustrate our bounds numerically in a simple example.