Update scores.rst

Author: SvenKlaassen

doc/guide/scores.rst

@@ -330,7 +330,7 @@ implements the score function:
 .. math::
 
     \psi(W,\theta, \eta) 
-    :=\; &-\theta + \left(\frac{D}{\mathbb{E}_n[D]} - \frac{\frac{m(X) (1-D)}{1-m(X)}}{\mathbb{E}_n\left[\frac{m(X) (1-D)}{1-m(X)}\right]}\right)\left(Y_1 - Y_0 -g(0,X)\right)
+    :=\; &-\theta + \left(\frac{D}{\mathbb{E}_n[D]} - \frac{1-D}{\mathbb{E}_n[1-D]}\right)\left(Y_1 - Y_0 -g(0,X)\right)
 
     &+ \left(1 - \frac{D}{\mathbb{E}_n[D]}\right) \left(g(1,X) - g(0,X)\right)
 
@@ -342,32 +342,30 @@ where the components of the linear score are
 
     \psi_a(W; \eta) \;=  &- 1,
 
-    \psi_b(W; \eta) \;= &\left(\frac{D}{\mathbb{E}_n[D]} - \frac{\frac{m(X) (1-D)}{1-m(X)}}{\mathbb{E}_n\left[\frac{m(X) (1-D)}{1-m(X)}\right]}\right)\left(Y_1 - Y_0 -g(0,X)\right)
+    \psi_b(W; \eta) \;= &\left(\frac{D}{\mathbb{E}_n[D]} - \frac{1-D}{\mathbb{E}_n[1-D]}\right)\left(Y_1 - Y_0 -g(0,X)\right)
 
     &+  \left(1 - \frac{D}{\mathbb{E}_n[D]}\right) \left(g(1,X) - g(0,X)\right)
 
-and the nuisance elements :math:`\eta=(g, m)` are defined as
+and the nuisance elements :math:`\eta=(g)` are defined as
 
 .. math::
 
     g_{0}(0, X) &= \mathbb{E}[Y_1 - Y_0|D=0, X]
 
     g_{0}(1, X) &= \mathbb{E}[Y_1 - Y_0|D=1, X]
 
-    m_0(X) &= P(D=1|X).
-
 Analogously, if ``in_sample_normalization='False'``,  the score is set to
 
 .. math::
 
     \psi(W,\theta, \eta) 
-    :=\; &-\theta +  \frac{D - m(X)}{p(1-m(X))}\left(Y_1 - Y_0 -g(0,X)\right)
+    :=\; &-\theta +  \frac{D - p}{p(1-p)}\left(Y_1 - Y_0 -g(0,X)\right)
 
     &+ \left(1 - \frac{D}{p}\right) \left(g(1,X) - g(0,X)\right)
 
     =\; &\psi_a(W; \eta) \theta + \psi_b(W; \eta)
 
-with :math:`\eta=(g, m, p)`, where :math:`p_0 = \mathbb{E}[D]` is estimated on the cross-fitting folds.
+with :math:`\eta=(g, p)`, where :math:`p_0 = \mathbb{E}[D]` is estimated on the cross-fitting folds.
 Remark that this will result in the same score, but just uses slightly different normalization.
 
 Difference-in-Differences for repeated cross-sections
@@ -408,13 +406,11 @@ where the components of the linear score are
 
     & + \frac{m(X) (1-D)(1-T)}{1-m(X)} \mathbb{E}_n\left[\frac{m(X) (1-D)(1-T)}{1-m(X)}\right]^{-1} (Y-g(0,0,X))
 
-and the nuisance elements :math:`\eta=(g, m)` are defined as
+and the nuisance elements :math:`\eta=(g)` are defined as
 
 .. math::
 
-    g_{0}(d, t, X) &= \mathbb{E}[Y|D=d, T=t, X]
-
-    m_0(X) &= P(D=1|X).
+    g_{0}(d, t, X) &= \mathbb{E}[Y|D=d, T=t, X].
 
 If ``in_sample_normalization='False'``, the score is set to
 
@@ -432,7 +428,7 @@ If ``in_sample_normalization='False'``, the score is set to
 
     =\; &\psi_a(W; \eta) \theta + \psi_b(W; \eta)
 
-with :math:`\eta=(g, m, p, \lambda)`, where :math:`p_0 = \mathbb{E}[D]` and :math:`\lambda_0 = \mathbb{E}[T]` are estimated on the cross-fitting folds.
+with :math:`\eta=(g, p, \lambda)`, where :math:`p_0 = \mathbb{E}[D]` and :math:`\lambda_0 = \mathbb{E}[T]` are estimated on the cross-fitting folds.
 Remark that this will result in the same score, but just uses slightly different normalization.
 
 ``score='experimental'`` assumes that the treatment probability is independent of the covariates :math:`X` and