The original version of this article unfortunately contained errors, which have been corrected with this erratum. In the paper, the semiparametric efficiency bound in Theorem 2 should be

$${\sum }_{\text{semi}}=\mathbb{E}\left\{\frac{\mathbb{V}[Y(1)|\mathit{X}]}{\pi (\mathit{X})}+\frac{\mathbb{V}[Y(0)|\mathit{X}]}{1-\pi (\mathit{X})}+{\left[{\mu }_1(\mathit{X})-{\mu }_0(\mathit{X})-{\tau }_{\mathrm{ATE}}\right]}^2\right\}..$$

This aligns with efficiency bound targeted in the proof within the online supplement.

Second, in equation (26), there is an errant $q_i$ included into the right-hand side of the second constraint that should be removed.

Finally, the description of the hdCBPS in Section 5.2 requires clarification. The itemized entry should instead state “An augmented version of CBPS that extends (34) by using regularized regression techniques to find debiased estimates of the potential outcome means.” The new wording better reflects the hdCBPS method versus the original description.

中文翻译：

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“使用布雷格曼距离的协变量平衡框架”勘误表

不幸的是，本文的原始版本包含错误，这些错误已通过此勘误进行更正。在本文中，定理2中的半参数效率界应为

$${\Sigma }_{\text{半}}=\mathbb{乙}\left\{\frac{\mathbb{V}[是（1）|\mathit{X}]}{\pi （\mathit{X}）}+\frac{\mathbb{V}[是（0）|\mathit{X}]}{1-\pi （\mathit{X}）}+{\left[{\mu }_1（\mathit{X}）-{\mu }_0（\mathit{X}）-{\tau }_{\mathrm{吃}}\right]}^2\right\}。。$$

这与在线补充证明中的目标效率界限一致。

其次，式（26）中存在一个错误$q_{我}$包含在应删除的第二个约束的右侧。

最后，5.2 节中 hdCBPS 的描述需要澄清。逐项条目应改为声明“CBPS 的增强版本，通过使用正则化回归技术来查找潜在结果均值的去偏估计来扩展 (34)”。与原始描述相比，新措辞更好地反映了 hdCBPS 方法。