Non-asymptotic confidence intervals for MCMC in practice

Understand

Using concentration inequalities, we give non-asymptotic confidence intervals for estimates obtained by Markov chain Monte Carlo (MCMC) simulations, when using the approximation $\mathbb{E}_{\pi} f\approx (1/(N-t_0))\cdot \sum_{i=t_0+1}^N f(X_i)$.

• To allow the application of non-asymptotic error bounds in practice, here we state bounds formulated in terms of the spectral properties of the chain and the properties of $f$ and propose estimators of the parameters appearing in the bounds, including the spectral gap, mixing time, and asymptotic variance.
• We introduce a method for setting the burn-in time and the initial distribution that is theoretically well-founded and yet is relatively simple to apply.
• We also investigate the estimation of $\mathbb{E}_{\pi}f$ via subsampling and by using parallel runs instead of a single run.