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无编辑摘要
* Any score that is 'finer' than the propensity score is a balancing score (i.e.: <math>e(X)=f(b(X))</math> for some function ''f''). The propensity score is the coarsest balancing score function, as it takes a (possibly) multidimensional object (''X''<sub>''i''</sub>) and transforms it into one dimension (although others, obviously, also exist), while <math>b(X)=X</math> is the finest one.
* Any score that is 'finer' than the propensity score is a balancing score (i.e.: <math>e(X)=f(b(X))</math> for some function ''f''). The propensity score is the coarsest balancing score function, as it takes a (possibly) multidimensional object (''X''<sub>''i''</sub>) and transforms it into one dimension (although others, obviously, also exist), while <math>b(X)=X</math> is the finest one.
* 任何比倾向性评分更“精细”的得分都是平衡得分（即：对于函数''f''，<math>e(X)=f(b(X))</math>）。倾向性评分是最粗粒度的平衡得分函数，因为它把一个（可能是）多维的对象（''X''<sub>''i''</sub>）转换成一维（尽管其他维度显然也存在），而<math>b(X)=X</math>则是最细粒度的平衡得分函数。
* 任何比倾向性评分更“精细”的得分都是平衡得分（即：对于函数''f''，<math>e(X)=f(b(X))</math>）。倾向性评分是最粗粒度的平衡得分函数，因为它把一个（可能是）多维的对象（''X''<sub>''i''</sub>）转换成只有一维（尽管其他维度显然也存在），而<math>b(X)=X</math>则是最细粒度的平衡得分函数（保留全部维度）。
* If treatment assignment is strongly ignorable given ''X'' then:
* If treatment assignment is strongly ignorable given ''X'' then:
If we think of the value of Z as a parameter of the population that impacts the distribution of X then the balancing score serves as a sufficient statistic for Z. Furthermore, the above theorems indicate that the propensity score is a minimal sufficient statistic if thinking of Z as a parameter of X. Lastly, if treatment assignment Z is strongly ignorable given X then the propensity score is a minimal sufficient statistic for the joint distribution of {\displaystyle (r_{0},r_{1})}{\displaystyle (r_{0},r_{1})}.
If we think of the value of Z as a parameter of the population that impacts the distribution of X then the balancing score serves as a sufficient statistic for Z. Furthermore, the above theorems indicate that the propensity score is a minimal sufficient statistic if thinking of Z as a parameter of X. Lastly, if treatment assignment Z is strongly ignorable given X then the propensity score is a minimal sufficient statistic for the joint distribution of {\displaystyle (r_{0},r_{1})}{\displaystyle (r_{0},r_{1})}.
如果我们把''Z''的值想成影响''X''分布的群体参数，则平衡得分充当了''Z''的充分统计量。进一步，上述定理指出，如果把''Z''视为''X''的参数，则倾向性评分就是最小充分统计量。最后，给定''X''，如果''Z''是强可忽略的，则倾向性评分是<math>(r_0, r_1)</math>联合分布的最小统计量。
如果我们把''Z''的值想成影响''X''分布的群体参数，则平衡得分充当了''Z''的充分统计量。进一步，上述定理指出，如果把''Z''视为''X''的参数，则倾向性评分就是最小充分统计量。最后，给定''X''，如果''Z''是强可忽略的，则倾向性评分是<math>(r_0, r_1)</math>联合分布的最小充分统计量。