MultinomialLogit

Multiclass logistic regression

from _api_doc_utils import *

1 Where it fits

Group: Regression

MultinomialLogit generalizes binary logit to \(K\) classes with softmax probabilities:

\[ \Pr(Y_i=k\mid X_i=x_i)=\frac{\exp(\alpha_k+x_i'\beta_k)}{\sum_\ell \exp(\alpha_\ell+x_i'\beta_\ell)}. \]

The summary identifies the last sorted class as reference_class and reports identifiable class-versus-reference coefficient contrasts. Fisher-information standard errors are available only for alpha=0.

2 Python API

Constructor: cm.MultinomialLogit

Use integer class labels in fit(x, y_int32). predict(x) returns class labels. summary() returns contrast rows aligned with class_labels; penalized fits mark inference unavailable and omit se/vcov.

print(inspect.signature(cm.MultinomialLogit))

(alpha=0.0, max_iterations=100, gradient_tolerance=0.0001)

cls = cm.MultinomialLogit
display(HTML(html_table(["Public method"], public_methods(cls))))

Public method
bootstrap(self, /, n_bootstrap, seed=None)
fit(self, /, x, y)
predict(self, /, x)
predict_label(self, /, x)
predict_lin(self, /, x)
summary(self, /)

3 Minimal example

rng = np.random.default_rng(6)
x = rng.normal(size=(240, 2))
logits = x @ np.array([[0.6, -0.3], [-0.4, 0.5], [0.2, 0.2]]).T + np.array([0.1, -0.2, 0.0])
p = np.exp(logits - logits.max(axis=1, keepdims=True))
p = p / p.sum(axis=1, keepdims=True)
y = np.array([rng.choice(3, p=row) for row in p], dtype=np.int32)
model = cm.MultinomialLogit(max_iterations=200)
model.fit(x, y)
print(model.summary()['coef'])
print(model.predict(x[:5]))

[[ 0.25831465  0.53237638 -0.55020945]
 [-0.15395855 -0.36374717  0.31279086]]
[[0.29713333 0.35470927 0.3481574 ]
 [0.10437553 0.60469433 0.29093014]
 [0.35652526 0.30570265 0.33777209]
 [0.27898794 0.37429093 0.34672113]
 [0.36790269 0.30230945 0.32978787]]

4 summary() contract

The table below is generated by fitting the live class in this repository and then inspecting summary(). Shapes are shown because most values are plain NumPy arrays or scalars.

rng = np.random.default_rng(106)
x = rng.normal(size=(100, 2))
logits = x @ np.array([[0.6, -0.3], [-0.4, 0.5], [0.2, 0.2]]).T
p = np.exp(logits - logits.max(1, keepdims=True))
p = p / p.sum(1, keepdims=True)
y = np.array([rng.choice(3, p=row) for row in p], dtype=np.int32)
model = cm.MultinomialLogit(max_iterations=200)
model.fit(x, y)
summary = model.summary()
display(HTML(html_table(["summary() key", "shape"], summary_shape_rows(summary))))

summary() key	shape
coef	(2, 3)
class_labels	(2,)
reference_class	()
penalty	()
inference_available	()
se	(2, 3)
vcov	(6, 6)