Not able to interpret decision tree when using class_weights

Question

I'm working with an imbalanced dataset. I'm using a decision tree (scikit-learn) to build a model.
For explaining my problem I've taken iris dataset.

When I'm setting class_weight=None, I understood how the tree is assigning the probability scores when I use predict_proba.
When I'm setting class_weight='balanced', I know its using target value to calculate class weights but I'm not able to understand how the tree is assigning the probability scores.

import sklearn.datasets as datasets
import pandas as pd
import numpy as np
from sklearn.tree import DecisionTreeClassifier
from sklearn.model_selection import train_test_split

from sklearn.externals.six import StringIO  
from IPython.display import Image  
from sklearn.tree import export_graphviz
import pydotplus

iris=datasets.load_iris()
df=pd.DataFrame(iris.data, columns=iris.feature_names)
y=iris.target

X_train, X_test, y_train, y_test = train_test_split(df, y, test_size=0.33, random_state=1)

# class_weight=None
dtree=DecisionTreeClassifier(max_depth=2)
dtree.fit(X_train,y_train)

dot_data = StringIO()
export_graphviz(dtree, out_file=dot_data, filled=True, rounded=True, special_characters=True, feature_names=X_train.columns)
graph = pydotplus.graph_from_dot_data(dot_data.getvalue())  
Image(graph.create_png()) # I use jupyter-notebook for visualizing the image

# printing unique probabilities in each class
probas = dtree.predict_proba(X_train)
print(np.unique(probas[:,0]))
print(np.unique(probas[:,1]))
print(np.unique(probas[:,2]))

# ratio for calculating probabilities
print(0/33, 0/34, 33/33)
print(0/33, 1/34, 30/33)
print(0/33, 3/33, 33/34)

The probabilities assigned by the tree and my ratios (determined by looking at tree image) are matching.

When I use the option class_weights='balanced'. I get the below tree.

# class_weight='balanced' 
dtree_balanced=DecisionTreeClassifier(max_depth=2, class_weight='balanced')
dtree_balanced.fit(X_train,y_train)

dot_data = StringIO()
export_graphviz(dtree_balanced, out_file=dot_data,filled=True, rounded=True, special_characters=True, feature_names=X_train.columns)
graph = pydotplus.graph_from_dot_data(dot_data.getvalue())  
Image(graph.create_png())

I'm printing unique probabilities using below code

probas = dtree_balanced.predict_proba(X_train)
print(np.unique(probas[:,0]))
print(np.unique(probas[:,1]))
print(np.unique(probas[:,2]))

I'm not able to understand (come-up with a formula) how the tree is assigning these probabilities.

Answer 1

We should consider two points. First, class_weight='balanced' does not change the actual number of samples in a class, only the weight of class $w_{c_i}$ is changed. Second, the [un-normalized] probability of class $c_i$ in each node is calculated as

$w_{c_i}$ x (number of samples from $c_i$ in that node / size of $c_i$)

For example, in balanced mode, the [un-normalized] probability of $c_3$ in the green leaf is calculated as

$33.\bar{3}\% \times (3 / 36) ≈ 2.778\%$

compared to $36\% \times (3 / 36) = 3\%$ in unbalanced mode.

The probability (normalized) in balanced mode would be:

$100 \times 2.778/(2.778+32.258) \% = 7.9289\%$

Remark. The word "probability" is not applicable to each isolated node except for the root node. This is the un-normalized version of the probability used to classify a data point inside a leaf, though the normalization is not required for comparison. However, the notion is applicable to the aggregate of nodes at the same level and the leaves from upper levels (i.e. set of all samples).