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 } {\rm  )}(}{\cal  {\cal   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%DIFDELCMD < \cal O%%% (\frac{TR^{2}{\varepsilon^{2}}) steps, when given as inputan (often loose) upper-bound T on the relaxation time \tau}  _{ %DIFDELCMD < \rm %%% rel } . When an upper-bound V on the stationary variance v_{\pi \doteq \mathbb{V}_{\pi}[f] is known, }%DIFDELCMD < \tilde{\cal O}\bigl(%%% \frac{TR{\varepsilon}+\frac{TV}{\varepsilon^{2}}}%DIFDELCMD < \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 ,  DynaMITE , and requires no bound on v_{\pi}. Receiving only an upper-bound }{\cal   on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{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\SUBSCRIPT}%DIFDELCMD < {%%%  \rm  . Thus we depend minimally on the tightness of }{\cal  , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to  is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal   to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm  )}(}{\cal  . Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal  {\rm  })}(}{\cal   = 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 }%DIFDELCMD < \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}  .
_{\pi}[f] of a } 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 %DIFDELCMD < \cal O%%% }{\varepsilon^{2}}) steps, when given as inputan (often loose) upper-bound T on the relaxation time \tau} receive, as input, upper-bounds on \tau _{ %DIFDELCMD < \rm %%% mix } or \tau_{rel the stationary variance \doteq \mathbb{V}_{\pi}[f] is known, }%DIFDELCMD < \tilde{\cal O}\bigl(%%% {\varepsilon}+\frac{TV}{\varepsilon^{2}}}%DIFDELCMD < \bigr) %%% , 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 %DIFDELCMD < \tilde{\cal O}%%% of f in\tilde{\cal{O}} \bigl( {\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + v_{\tau\rm rel}}{\varepsilon^{2}}} \frac{\tau_{rel\cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without , the variance of f, or the trace variance v_{T}. When \epsilon is small, } a priori bounds on the stationary variance v , thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%% _{\pi \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\SUBSCRIPT}{ 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} \leq 2v_{\pi}, thus the worst-case complexity of DynaMITE is }%DIFDELCMD < \tilde{\cal O}%%% {\varepsilon} +\frac{\tau_{\rm rel} v_{\pi}}{\varepsilon^{2}}), improving the dependence of classical methods on the loose bounds T and V} (f)), thus DynaMITE outperforms prior methods in these cases .
D
The Markov-Chain Monte-Carlo (MCMC) method has been used widely in the literature for various applications, in particular estimating the expectation \mathbb{E
A
We introduce a novel statistical measure for MCMC-mean estimation, the inter-trace variance
A
trv
A
^{(\tau_{rel
A
M
A
,f), which depends on a Markov chain
A
M
A
and a
R
\Omega
S
D
over a distribution \pi on \Omega (a. k.a
R
mean-estimation), to within \varepsilon additive error (w.h.p.). Letting R \doteq b-a, standard variance-agnostic
The inter-trace variance can be efficiently estimated from observed data and leads to a more efficient MCMC-mean estimator. Prior
D
run the chain for
D
(\frac{TR^{2
A
receive, as input, upper-bounds on \tau
R
rel
mix
R
. When an upper-bound V on
or \tau_{rel
D
v_{\pi
D
\frac{TR
D
steps suffice. We introduce the DYNAmic Mcmc Inter-Trace variance Estimation (DynaMITE) algorithm for mean-estimation. We define the inter-trace variance v_{T
A
and their performance is highly dependent to the sharpness of these bounds. In contrast, we introduce
A
, which dynamically adjusts the sample size, it is less sensitive to the looseness of input upper-bounds on \tau_{rel
A
T
A
_{rel
D
within \varepsilon additive error within
A
of f in
D
\frac{TR
D
\frac{\tau_{\rm rel
A
\frac{\tau_{rel
D
a priori
D
bounds on v_{\pi
A
a priori bounds on
R
dominating term is \tau_{\rm rel
stationary variance
D
_{\tau\rm rel
A
_{\pi
A
trv
A
^{(\tau rel)
A
T
A
_{mix
A
^{(\tau{rel
A
0. Note that bounding \tau_{\rm rel
A
T
A
_{rel
R
and v_{\tau\rm rel
, simply by exploiting properties of the inter-trace variance. To compare our method to known variance-aware bounds, we show
A
trv
A
^{(\tau{rel
A
M
A
,f) \leq v_{\pi
A
M
A
's traces, we have
A
trv
A
^{({\tau{rel
A
M
A
,f)
R
o(v
o(v
D
), and we identify two cases to demonstrate it. Furthermore,it always holds that v_{\tau\rm rel
D
(\frac{TR
A
(f)), thus DynaMITE outperforms prior methods in these cases