The Markov-Chain Monte-Carlo (MCMC) method has been used widely in the
literature for various applications, in particular estimating the expectation
$\mathbb{E}_{\pi}[f]$ of a function $f:\Omega\to [a,b]$ over a distribution
$\pi$ on $\Omega$ (a.k.a. mean-estimation), to within $\varepsilon$ additive
error (w.h.p.). Letting $R \doteq b-a$, standard variance-agnostic MCMC
mean-estimators run the chain for $\tilde{\cal
O}(\frac{TR^{2}}{\varepsilon^{2}})$ steps, when given as input an (often loose)
upper-bound $T$ on the relaxation time $\tau_{\rm rel}$. When an upper-bound
$V$ on the stationary variance $v_{\pi} \doteq \mathbb{V}_{\pi}[f]$ is known,
$\tilde{\cal O}\bigl(\frac{TR}{\varepsilon}+\frac{TV}{\varepsilon^{2}}\bigr)$
steps suffice. We introduce the DYNAmic Mcmc Inter-Trace variance Estimation
(DynaMITE) algorithm for mean-estimation. We define the inter-trace variance
$v_{T}$ for any trace length $T$, and show that w.h.p., DynaMITE estimates the
mean within $\varepsilon$ additive error within $\tilde{\cal
O}\bigl(\frac{TR}{\varepsilon} + \frac{\tau_{\rm rel} v_{\tau\rm
rel}}{\varepsilon^{2}}\bigr)$ steps, without a priori bounds on $v_{\pi}$,
the variance of $f$, or the trace variance $v_{T}$.
When $\epsilon$ is small, the dominating term is $\tau_{\rm rel} v_{\tau\rm
rel}$, thus the complexity of DynaMITE principally depends on the a priori
unknown $\tau_{\rm rel}$ and $v_{\tau\rm rel}$. We believe in many situations
$v_{T}=o(v_{\pi})$, and we identify two cases to demonstrate it. Furthermore,
it always holds that $v_{\tau\rm rel} \leq 2v_{\pi}$, thus the worst-case
complexity of DynaMITE is $\tilde{\cal O}(\frac{TR}{\varepsilon}
+\frac{\tau_{\rm rel} v_{\pi}}{\varepsilon^{2}})$, improving the dependence of
classical methods on the loose bounds $T$ and $V$.