使用MCMC跟踪方差方法使均值估计更有效：DynaMITE
Making mean-estimation more efficient using an MCMC trace variance approach: DynaMITE
摘要：
我们介绍了一种新的统计测量MCMC均值估计，道间方差${\rm trv}^{(τ_{rel})}({\cal M},f)$，它取决于一个马尔可夫链${\cal M}$和一个功能$f:S\to [a,b]$。道间方差可以有效地估计从观测数据，并导致更有效的MCMC均值估计。先前MCMC均值估计器接收$τ_{mix}$或$τ_{rel}$上的上界作为输入，并且通常还接收平稳方差，并且它们的性能高度依赖于这些界的锐度。相比之下，我们引入了动态调整样本大小的DynaMITE，它对$τ_{rel}$上的输入上限的宽松性不太敏感，并且不需要对$v_π$进行限制。仅接收$τ_{rel}$上的上限${\cal T}_{rel}$时，DynaMITE以$\tilde{\cal{O}}\bigl(\smash{\frac{{\cal T}_{rel} R}{\varepsilon}}+\frac{τ_{rel}\cdot {\rm trv}^{(τ{rel})}}{\varepsilon^{2}}\bigr)$为步长估计$f$的均值，而无需对平稳方差$v_π$或迹间方差${\rm trv}^{(τrel)}$进行先验边界。因此，我们最低限度地依赖于${\cal T}_{mix}$的紧密性，因为复杂性由$τ_{rel}\rm{trv}^{(τ{rel})}$作为$\varepsilon \to 0$主导。请注意，已知绑定$τ_{\rm rel}$非常困难，但是，DynaMITE能够简单地利用迹间方差的特性，将其对${\cal T}_{rel}$的主要依赖性降低到$τ_{rel}$。为了将我们的方法与已知的方差感知界限进行比较，我们展示了${\rm trv}^{(τ{rel})}({\cal M},f) \leq v_π$。此外，我们表明，当$f$的图像分布（半）对称的${\cal M}$的痕迹，我们有${\rm trv}^{({τ{rel}})}({\cal M},f)=o(v_π(f))$，因此DynaMITE优于以前的方法在这些情况下。
英文摘要：
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.
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