Making mean-estimation more efficient using an MCMC trace variance approach: DynaMITE

Abstract

We introduce a novel statistical measure for MCMC-mean estimation, the inter-trace variance ${\rm trv}^{(\tau_{rel})}({\cal M},f)$, which depends on a Markov chain ${\cal M}$ and a function $f:S\to [a,b]$. The inter-trace variance can be efficiently estimated from observed data and leads to a more efficient MCMC-mean estimator. Prior MCMC mean-estimators receive, as input, upper-bounds on $\tau_{mix}$ or $\tau_{rel}$, and often also the stationary variance, and their performance is highly dependent to the sharpness of these bounds. In contrast, we introduce DynaMITE, which dynamically adjusts the sample size, it is less sensitive to the looseness of input upper-bounds on $\tau_{rel}$, and requires no bound on $v_{\pi}$. Receiving only an upper-bound ${\cal T}_{rel}$ on $\tau_{rel}$, DynaMITE estimates the mean of $f$ in $\tilde{\cal{O}}\bigl(\smash{\frac{{\cal T}_{rel} R}{\varepsilon}}+\frac{\tau_{rel}\cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}\bigr)$ steps, without a priori bounds on the stationary variance $v_{\pi}$ or the inter-trace variance ${\rm trv}^{(\tau rel)}$. Thus we depend minimally on the tightness of ${\cal T}_{mix}$, as the complexity is dominated by $\tau_{rel}\rm{trv}^{(\tau{rel})}$ as $\varepsilon \to 0$. Note that bounding $\tau_{\rm rel}$ is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on ${\cal T}_{rel}$ to $\tau_{rel}$, simply by exploiting properties of the inter-trace variance. To compare our method to known variance-aware bounds, we show ${\rm trv}^{(\tau{rel})}({\cal M},f) \leq v_{\pi}$. Furthermore, we show when $f$'s image is distributed (semi)symmetrically on ${\cal M}$'s traces, we have ${\rm trv}^{({\tau{rel}})}({\cal M},f)=o(v_{\pi}(f))$, thus DynaMITE outperforms prior methods in these cases.