DOC Elaborate on the criterion used for poisson regression using decision trees. (scikit-learn#29230)

Author: m-maggi

doc/modules/tree.rst

@@ -552,17 +552,18 @@ Mean Squared Error:
 
     H(Q_m) = \frac{1}{n_m} \sum_{y \in Q_m} (y - \bar{y}_m)^2
 
-Half Poisson deviance:
+Mean Poisson deviance:
 
 .. math::
 
-    H(Q_m) = \frac{1}{n_m} \sum_{y \in Q_m} (y \log\frac{y}{\bar{y}_m}
+    H(Q_m) = \frac{2}{n_m} \sum_{y \in Q_m} (y \log\frac{y}{\bar{y}_m}
     - y + \bar{y}_m)
 
 Setting `criterion="poisson"` might be a good choice if your target is a count
 or a frequency (count per some unit). In any case, :math:`y >= 0` is a
 necessary condition to use this criterion. Note that it fits much slower than
-the MSE criterion.
+the MSE criterion. For performance reasons the actual implementation minimizes
+the half mean poisson deviance, i.e. the mean poisson deviance divided by 2.
 
 Mean Absolute Error:
 

sklearn/tree/_classes.py

@@ -1098,7 +1098,7 @@ class DecisionTreeRegressor(RegressorMixin, BaseDecisionTree):
         mean squared error with Friedman's improvement score for potential
         splits, "absolute_error" for the mean absolute error, which minimizes
         the L1 loss using the median of each terminal node, and "poisson" which
-        uses reduction in Poisson deviance to find splits.
+        uses reduction in the half mean Poisson deviance to find splits.
 
         .. versionadded:: 0.18
            Mean Absolute Error (MAE) criterion.