二项和多项比例：精确估计和可靠的准确性评估
Binomial and Multinomial Proportions: Accurate Estimation and Reliable Assessment of Accuracy
摘要：
$σ_{P_o}$的错误估计，$P_o$中的\emph{uncertainty}来自用于二进制分类的2状态贝叶斯方程，显然是由$\hatσ_{p_i}$引起的，$\hatσ_{p_i}$是根据实验$b$ -bin直方图估计的基础pdf的不确定性。为了解决这一问题，比较了几个贝叶斯估计量对$(\hat{p}_i, \hatσ_{p_i})$的标称置信水平（$ξ$）和计算的覆盖值（$C$）之间的一致性。对于所有多项估计量，大$b$和$ p_i \gg \frac{1}{b}$的大$ξ$到$C$不一致性都出现，因为先验降低了低似然、高$p_i$值的权重。为了改善$ξ$与$C$的匹配，在更一般的先验pdf（$\mathcal{B}[α_0,(b-1)α_0;x]$）中将$(ξ-C)^2$相对于$α_0$最小化，以获得$(\hat{p_i})_{ξ\leftrightarrow C}$ .这改进了$b=2$的匹配，但是对于$b>2$，通过$(\hat{p_i})_{ξ\leftrightarrow C}$的$ξ$到$C$的匹配需要有效值“$b=2$“和重正化，这减少了$\hat{p}_i$到$p_i$的匹配.更好的$\hat{p}_i$到$p_i$匹配来自原始多项式估计量、新的离散域估计量$\hat{p}(n_i,N)$或早期的\emph{joint}估计量$(\hat{p_i})_{\bowtie}$，其将James-Stein收缩的所有估计值$p_i$共同调整为均值向量。最佳的同时$ξ$到$C$和$\hat{p}_i$到$p_i$匹配来自基础PDF的\emph{de-noising}初始估计.对于$b=100$、$N<12800$、去噪的$\hat{p}$需要$\approx 10\times$更少的观测值来实现与针对$\hat{p}(n_i,N)$、$(\hat{p_i})_{\bowtie}$或原始多项式$\hat{p}_i$所发现的$\hat{p}_i$到$p_i$匹配等效的$\hat{p}_i$到$p_i$匹配。对每种不同类型的初始估计进行去噪，在Monte-Carlo测试中产生了类似的高精度。
英文摘要：
Misestimates of $σ_{P_o}$, the \emph{uncertainty} in $P_o$ from a 2-state Bayes equation used for binary classification, apparently arose from $\hatσ_{p_i}$, the uncertainty in underlying pdfs estimated from experimental $b$-bin histograms. To address this, several Bayesian estimator pairs $(\hat{p}_i, \hatσ_{p_i})$ were compared for agreement between nominal confidence level ($ξ$) and calculated coverage values ($C$). Large $ξ$-to-$C$ inconsistency for large $b$ and $ p_i \gg \frac{1}{b}$ arises for all multinomial estimators since priors downweight low likelihood, high $p_i$ values. To improve $ξ$-to-$C$ matching, $(ξ-C)^2$ was minimized against $α_0$ in a more general prior pdf ($\mathcal{B}[α_0,(b-1)α_0;x]$) to obtain $(\hat{p_i})_{ξ\leftrightarrow C}$. This improved matching for $b=2$, but for $b>2$, $ξ$-to-$C$ matching by $(\hat{p_i})_{ξ\leftrightarrow C}$ required an effective value "$b=2$" and renormalization, and this reduced $\hat{p}_i$-to-$p_i$ matching. Better $\hat{p}_i$-to-$p_i$ matching came from the original multinomial estimators, a new discrete-domain estimator $\hat{p}(n_i,N)$, or an earlier \emph{joint} estimator, $(\hat{p_i})_{\bowtie}$ that co-adjusted all estimates $p_i$ for James-Stein shrinkage to a mean vector. Best simultaneous $ξ$-to-$C$ and $\hat{p}_i$-to-$p_i$ matching came by \emph{de-noising} initial estimates of underlying pdfs. For $b=100$, $N<12800$, de-noised $\hat{p}$ needed $\approx 10\times$ fewer observations to achieve $\hat{p}_i$-to-$p_i$ matching equivalent to that found for $\hat{p}(n_i,N)$, $(\hat{p_i})_{\bowtie}$ or the original multinomial $\hat{p}_i$. De-noising each different type of initial estimate yielded similarly high accuracy in Monte-Carlo tests.
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