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

We introduce a novel statistical measure for MCMC-mean estimation, the inter-trace variance rm trv^(τrel)(cal M,f), which depends on a Markov chain cal M and a function f:S→ [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 τmix or τrel, 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 τrel, and requires no bound on vπ. Receiving only an upper-bound cal Trel on τrel, DynaMITE estimates the mean of f in tildecalObigl(smashfraccal Trel Rvarepsilon+fracτrel· rm trv^(τrel)varepsilon2bigr) steps, without a priori bounds on the stationary variance vπ or the inter-trace variance rm trv(τ rel). Thus we depend minimally on the tightness of cal Tmix, as the complexity is dominated by τrelrmtrv^(τrel) as varepsilon → 0. Note that bounding τrm rel is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on cal Trel to τrel, simply by exploiting properties of the inter-trace variance. To compare our method to known variance-aware bounds, we show rm trv^(τrel)(cal M,f) ≤ vπ. Furthermore, we show when f's image is distributed (semi)symmetrically on cal M's traces, we have rm trv^(τrel)(cal M,f)=o(vπ(f)), thus DynaMITE outperforms prior methods in these cases.