论文标题
使用MCMC痕量方差方法使均值估计更有效：炸药
Making mean-estimation more efficient using an MCMC trace variance approach: DynaMITE
论文作者
论文摘要
我们介绍了一种用于MCMC均值估计的新颖统计度量，Trace差异$ {\ rm trv}^{（τ_{rel}）}}（{\ cal m}，f），F）$，取决于Markov链$ {\ cal m} $ {\ cal m} $ and a function $ f：s \：s \ to [a f to to [\ cal m} $。可以从观察到的数据中有效估计跟踪方差，并导致更有效的MCMC均值估计器。在$τ_{mix} $或$τ_{rel} $上，以前的MCMC均值估计器作为输入接收到上限，并且通常也是固定的差异，并且它们的性能高度取决于这些边界的敏锐度。相比之下，我们引入了动态调整样本量的炸药，它对$τ_{rel} $对输入上限的松散性的敏感性不太敏感，并且不需要在$v_π$上绑定。 在$τ_{rel} $上仅接收上限$ {\ cal t} _ {rel} $，Dynamite估算$ f $ in $ \ tilde {\ cal {o}} \ bigl（ r} {\ varepsilon}}}+\ frac {τ_{rel} \ cdot {\ rm trv}^{（τ{rel}）}}}} {\ varepsilon^{2}}}} \ bigR spess，没有先前的$ v_ $ v_ $ v_ trv}^{（τrel）} $。因此，我们最大程度地依赖于$ {\ cal t} _ {mix} $的紧密度，因为复杂性由$τ_{rel} \ rm {rm {trv}^{（τ{rel}）} $作为$ \ varepsilon \ to varepsilon \ to $ \ varepsilon \ to 0 $。请注意，众所周知，$τ_ {\ rm rel} $非常困难，但是，炸药能够将其主要依赖的依赖性减少到$ {\ cal t} _ {rel} _ {rel} $至$τ_{rem} $，简单地通过利用跨传递差异的属性来减少。为了将我们的方法与已知的方差感知界限进行比较，我们显示$ {\ rm trv}^{（τ{rel}）}}（{\ cal m}，f），f）\ leqv_π$。此外，我们在$ {\ cal m} $ traces上对$ f $的图像进行了分布（半度）的范围时，我们有$ {\ rm trv}^{（{°{τ{rel}}}}}}（{\ cal m}，{\ cal m}，f），f），f）= o（v_π（v_πf）
We introduce a novel statistical measure for MCMC-mean estimation, the inter-trace variance ${\rm trv}^{(τ_{rel})}({\cal M},f)$, 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 $τ_{mix}$ or $τ_{rel}$, 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 $τ_{rel}$, and requires no bound on $v_π$. Receiving only an upper-bound ${\cal T}_{rel}$ on $τ_{rel}$, DynaMITE estimates the mean of $f$ in $\tilde{\cal{O}}\bigl(\smash{\frac{{\cal T}_{rel} R}{\varepsilon}}+\frac{τ_{rel}\cdot {\rm trv}^{(τ{rel})}}{\varepsilon^{2}}\bigr)$ steps, without a priori bounds on the stationary variance $v_π$ or the inter-trace variance ${\rm trv}^{(τrel)}$. Thus we depend minimally on the tightness of ${\cal T}_{mix}$, as the complexity is dominated by $τ_{rel}\rm{trv}^{(τ{rel})}$ as $\varepsilon \to 0$. Note that bounding $τ_{\rm rel}$ is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on ${\cal T}_{rel}$ to $τ_{rel}$, simply by exploiting properties of the inter-trace variance. To compare our method to known variance-aware bounds, we show ${\rm trv}^{(τ{rel})}({\cal M},f) \leq v_π$. Furthermore, we show when $f$'s image is distributed (semi)symmetrically on ${\cal M}$'s traces, we have ${\rm trv}^{({τ{rel}})}({\cal M},f)=o(v_π(f))$, thus DynaMITE outperforms prior methods in these cases.
