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

The Markov-Chain Monte-Carlo (MCMC) method has been used widely in the literature for various applications, in particular estimating the expectation πΌ_Ο[f] of a function f:Ξ©β [a,b] over a distribution Ο on Ξ© (a.k.a. mean-estimation), to within Ξ΅ additive error (w.h.p.). Letting R β b-a, standard variance-agnostic MCMC mean-estimators run the chain for Γ(TR^2/Ξ΅^2) steps, when given as input an (often loose) upper-bound T on the relaxation time Ο_ rel. When an upper-bound V on the stationary variance v_Οβπ_Ο[f] is known, Γ(TR/Ξ΅+TV/Ξ΅^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 for any trace length T, and show that w.h.p., DynaMITE estimates the mean within Ξ΅ additive error within Γ(TR/Ξ΅ + Ο_ rel v_Ο rel/Ξ΅^2) steps, without a priori bounds on v_Ο, the variance of f, or the trace variance v_T. When Ο΅ is small, the dominating term is Ο_ rel v_Ο rel, thus the complexity of DynaMITE principally depends on the a priori unknown Ο_ rel and v_Ο rel. We believe in many situations v_T=o(v_Ο), and we identify two cases to demonstrate it. Furthermore, it always holds that v_Ο relβ€ 2v_Ο, thus the worst-case complexity of DynaMITE is Γ(TR/Ξ΅ +Ο_ rel v_Ο/Ξ΅^2), improving the dependence of classical methods on the loose bounds T and V.

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