fixing math, harmonising different notations.

hazardous/_ipcw.py

@@ -144,8 +144,8 @@ def compute_censoring_survival_proba(self, times, X=None, ipcw_training=False):
 class AlternatingCensoringEstimator(KaplanMeierIPCW):
     r"""IPCW estimator for Debiased Gradient Boosting Incidence.
 
-    Predict :math:`\hat{G}(t | X = x) = p(C > t | X = x)` using
-    :math:`1/\hat{S}(t | X = x) = 1/p(T^* > t | X = x)` as IPCW.
+    Predict :math:`\hat{G}(t | X = x) = P(C > t | X = x)` using
+    :math:`1/\hat{S}(t | X = x) = 1/P(T^* > t | X = x)` as IPCW.
 
     TODO
     """
@@ -227,8 +227,8 @@ def compute_censoring_survival_proba(self, times, X=None, ipcw_training=False):
         for each :math:`k` competing event.
 
         The probabilities returned by the censoring estimator are
-        :math:`\hat{S}(t| X = x) = P(T^* > t | X = x)` and
-        :math:`\hat{G}(t) = P(C^* > t | X = x)`.
+        :math:`\hat{G}(t) = P(C > t | X = x)` and
+        :math:`1 - \hat{G}(t) = P(C \leq t | X = x)`.
 
         Parameters
         ----------

hazardous/_survival_boost.py

@@ -90,7 +90,7 @@ def draw(self, ipcw_training=False, X=None):
         sampled_time_horizons[hard_zero_indices] = 0.0
 
         if ipcw_training:
-            # During the training of the ICPW, we estimate G(t) = P(C > t)
+            # During the training of the ICPW, we estimate G(t) = P(C > t) using
             # 1 / S(t) = 1 / P(T^* > t) as sample weight. t is an arbitrary
             # time horizon.
             # Since 1 = P(C <= t) + P(C > t), our training target is the censoring
@@ -180,19 +180,19 @@ class SurvivalBoost(BaseEstimator, ClassifierMixin):
 
     .. math::
 
-        \hat{F}_k(t; x_i) \approx F_k(t; x_i) = \mathbb{P}(T \leq t, E=k | X=x_i)
+        \hat{F}_k(t; x_i) \approx F_k(t; x_i) = \mathbb{P}(T \leq t, \Delta=k | X=x_i)
 
     where :math:`T` is a random variable for the uncensored time to first event
-    and :math:`E` is a random variable over the :math:`[1, K]` domain for the
+    and :math:`\Delta` is a random variable over the :math:`[1, K]` domain for the
     (uncensored) event type, and :math:`x_i` is the feature vector of the
     :math:`i`-th observation.
 
     The (any event) Survival Function can be defined as:
 
     .. math::
 
-        S(t; x_i) = \mathbb{P}(T > t) = 1 - \mathbb{P}(T \leq t | X=x_i)
-        = 1 - \sum_{k=1}^K \mathbb{P}(T \leq t, E=k | X=x_i)
+        S(t; x_i) = \mathbb{P}(T > t | X=x_i) = 1 - \mathbb{P}(T \leq t | X=x_i)
+        = 1 - \sum_{k=1}^K \mathbb{P}(T \leq t, \Delta=k | X=x_i)
         = 1 - \sum_{k=1}^K F_k(t; x_i)
 
     Under the hood, this class randomly samples reference time horizons

hazardous/metrics/_brier_score.py

@@ -409,7 +409,7 @@ def brier_score_incidence(
 
     .. math::
 
-            \hat{F}_k(t | \mathbf{x}_i) \approx P(T_i \leq t, E_i = k |
+            \hat{F}_k(t | \mathbf{x}_i) \approx P(T_i \leq t, \Delta_i = k |
             \mathbf{x}_i)
 
     and :math:`\hat{\omega}_i(t)` are IPCW weigths based on the Kaplan-Meier