Before revision:  Letting R \doteq b-a, standard variance-agnostic MCMC mean-estimators run the chain for (\frac{TR^{2}{\varepsilon^{2}}) steps, when given as inputan (often loose) upper-bound T on the relaxation time \tau}  _{  rel } . When an upper-bound V on the stationary variance v_{\pi \doteq \mathbb{V}_{\pi}[f] is known, } \frac{TR{\varepsilon}+\frac{TV}{\varepsilon^{2}}} steps suffice. We introduce the DYNAmic Mcmc Inter-Trace variance Estimation (DynaMITE) algorithm for mean-estimation. We define the inter-trace variance v_{T ,  DynaMITE , and requires no bound on v_{\pi}.
After revision:  Letting R \doteq b-a, standard variance-agnostic MCMC mean-estimators run the chain for (\frac{TR^{2}{\varepsilon^{2}}) steps, when given as inputan (often loose) upper-bound T on the relaxation time \tau} _{  rel } . When an upper-bound V on the stationary variance v_{\pi \doteq \mathbb{V}_{\pi}[f] is known, } \frac{TR{\varepsilon}+\frac{TV}{\varepsilon^{2}}} , DynaMITE , and requires no bound on v_{\pi}.