Multinomial Logistic Regression

In multinomial logistic regression (MLR) the logistic function we saw in Recipe 15.1 is replaced with a softmax function:

$$P(y_i=k \mid X)={\frac {e^{\beta_{k}x_{i}}}{{\sum_{j=1}^{K}}e^{\beta_{j}x_{i}}}}$$

where $P(y_i=k \mid X)$ is the probability the $i$th observation’s target value, $y_i$, is class $k$, and $K$ is the total number of classes. One practical advantage of the MLR is that its predicted probabilities using the predict_proba method are more reliable (i.e. better calibrated).


# Load libraries
from sklearn.linear_model import LogisticRegression
from sklearn import datasets
from sklearn.preprocessing import StandardScaler

Load Iris Flower Data

# Load data
iris = datasets.load_iris()
X =
y =

Standardize Features

# Standarize features
scaler = StandardScaler()
X_std = scaler.fit_transform(X)

Create Multinomial Logistic Regression

# Create one-vs-rest logistic regression object
clf = LogisticRegression(random_state=0, multi_class='multinomial', solver='newton-cg')

Train Multinomial Logistic Regression

# Train model
model =, y)

Create Previously Unseen Observation

# Create new observation
new_observation = [[.5, .5, .5, .5]]

Predict Observation’s Class

# Predict class

View Probability Observation Is Each Class

# View predicted probabilities
array([[ 0.01944996,  0.74469584,  0.2358542 ]])