Multiclass Logistic

🎯 Purpose

This QuickRef covers multinomial (multiclass) logistic regression — a generalization of binary logistic used when your target variable has 3 or more unordered classes.

📦 1. When to Use¶

Condition	Use Multiclass Logistic
Target has ≥3 unordered categories	✅ Yes
Categories have no rank or order	✅ Yes
Binary target	❌ Use standard logistic
Ordered categories (low < med < high)	❌ Use ordinal logistic

Examples: Customer segments, product types, political affiliation

🧮 2. Model Logic (Softmax Function)¶

$$ P(Y = k \mid X) = \frac{\exp(X \cdot \beta_k)}{\sum_{j=1}^{K} \exp(X \cdot \beta_j)} $$

Each class k has its own coefficient vector β_k, compared to a baseline (reference) class.

🛠️ 3. Fitting the Model¶

from sklearn.linear_model import LogisticRegression
clf = LogisticRegression(multi_class='multinomial', solver='lbfgs')
clf.fit(X_train, y_train)

# statsmodels implementation
import statsmodels.api as sm
model = sm.MNLogit(y, sm.add_constant(X)).fit()

✔️ Use predict_proba() for class probabilities

📊 4. Output Interpretation¶

Output	Meaning
`coef_`	Log-odds of class k vs reference
`np.exp(coef_)`	Odds ratio for each class vs baseline
`predict_proba()`	Class probability distribution per row

⚠️ 5. Assumptions¶

Assumption	Notes
Independence of Irrelevant Alternatives (IIA)	Assumes relative odds between any two categories do not depend on the presence of others
No ordering	Categories must be nominal, not ordinal

✅ Modeling Checklist¶

[ ] Confirm target is nominal (unordered, ≥3 classes)
[ ] Model fit using multi_class='multinomial' and proper solver (e.g. lbfgs)
[ ] Probabilities interpreted using softmax logic
[ ] Log-odds and odds ratios clearly reported
[ ] IIA assumption acknowledged or tested (esp. in statsmodels)

💡 Tip¶

“Binary logistic picks yes/no. Multiclass logistic asks: which one?”