update sensitivity documentation

Author: SvenKlaassen

doc/guide/sensitivity.rst

@@ -113,7 +113,14 @@ As :math:`\sigma_0^2` and :math:`\nu_0^2` do not depend on the unobserved confou
 
 - ``cf_d``:math:`:=1 - \frac{\mathbb{E}\big[\alpha(W)^2\big]}{\mathbb{E}\big[\tilde{\alpha}(\tilde{W})^2\big]}` measures the proportion of residual variance in the Riesz representer :math:`\tilde{\alpha}(\tilde{W})` generated by the latent confounders :math:`A`
 
-For model-specific interpretations of ``cf_d``, see the corresponding chapters (e.g. :ref:`sensitivity_plr`).
+.. note::
+    - ``cf_y`` has the interpretation as the *nonparametric partial* :math:`R^2` *of* :math:`A` *with* :math:`Y` *given :math:`(D,X)`*
+    .. math:: 
+        
+        \frac{\textrm{Var}(\mathbb{E}[Y|D,X,A]) - \textrm{Var}(\mathbb{E}[Y|D,X])}{\textrm{Var}(Y)-\textrm{Var}(\mathbb{E}[Y|D,X])}
+
+    - For model-specific interpretations of ``cf_d`` or :math:`C_D^2`, see the corresponding chapters (e.g. :ref:`sensitivity_plr`).
+
 Consequently, for given values ``cf_y`` and ``cf_d``, we can create lower and upper bounds for target parameter :math:`\tilde{\theta}_0` of the form
 
 .. math::
@@ -282,6 +289,19 @@ Therefore,
 
 - ``cf_d``:math:`:=\frac{\mathbb{E}\big[\big(\mathbb{E}[D|X,A] - \mathbb{E}[D|X]\big)^2\big]}{\mathbb{E}\big[\big(D - \mathbb{E}[D|X]\big)^2\big]}` measures the proportion of residual variance in the treatment :math:`D` explained by the latent confounders :math:`A`
 
+.. note::
+    In the :ref:`plr-model`, both ``cf_y`` and ``cf_d`` can be interpreted as *nonparametric partial* :math:`R^2`
+
+    - ``cf_y`` has the interpretation as the *nonparametric partial* :math:`R^2` *of* :math:`A` *with* :math:`Y` *given* :math:`(D,X)`
+    .. math:: 
+        
+        \frac{\textrm{Var}(\mathbb{E}[Y|D,X,A]) - \textrm{Var}(\mathbb{E}[Y|D,X])}{\textrm{Var}(Y)-\textrm{Var}(\mathbb{E}[Y|D,X])}
+
+    - ``cf_d`` has the interpretation as the *nonparametric partial* :math:`R^2` *of* :math:`A` *with* :math:`D` *given* :math:`X`
+    .. math:: 
+        
+        \frac{\textrm{Var}(\mathbb{E}[D|X,A]) - \textrm{Var}(\mathbb{E}[D|X])}{\textrm{Var}(D)-\textrm{Var}(\mathbb{E}[D|X])}
+
 Using the partially linear regression model with ``score='partialling out'`` the ``nuisance_elements`` are implemented in the following form
 
 .. math::
@@ -326,6 +346,16 @@ This implies the following representations
 
     \alpha(W) &= \bigg(\frac{D}{m(X)} - \frac{1-D}{1-m(X)}\bigg)  \mathbb{E}[\omega(D,X)|X].
 
+.. note::
+    In the :ref:`irm-model` with the ATE, it holds
+    
+    .. math:: 
+
+        C_D^2= \frac{\mathbb{E}\Big[\big(P(D=1|X,A)(1-P(D=1|X,A))\big)^{-1}\Big]}{\mathbb{E}\Big[\big(P(D=1|X)(1-P(D=1|X))\big)^{-1}\Big]} - 1
+    
+    which is the *average gain in conditional precision to predict* :math:`D` *by using* :math:`A` *in addition to* :math:`X`.
+    This can be used to choose ``cf_d``:math:`:=\frac{C_D^2}{1 + C_D^2}`.
+
 The ``nuisance_elements`` are then computed with plug-in versions according to the general :ref:`sensitivity_implementation`. 
 For ``score='ATE'``, the weights are set to one