Maximum likelihood estimation chooses the parameter values that make the observed data most likely. In logistic regression, it adjusts β₀ and β₁ to find the most plausible curve for the observed outcomes. section 8.3
Unlike least squares, this approach asks which curve most likely produced the data. The fitting mathematics stays mostly behind the library interface. section 8.3
A practical workflow looks like this:
- Prepare predictor values as a column and pair them with binary outcomes. section 8.3
- Fit LogisticRegression with scikit-learn, then inspect the learned coefficient and intercept. section 8.3
- Use the fitted parameters to calculate or predict an outcome probability. section 8.3
Toddler meltdowns
Given hours without a nap and whether a meltdown occurred, maximum likelihood estimation fits a logistic curve. The resulting model predicts about a 0.9701 meltdown probability at 4.78 hours without a nap. section 8.3
import numpy as np
from sklearn.linear_model import LogisticRegression
x = np.array([0.1, 0.5, 0.7, 1.2, 1.2, 1.5, 1.6, 1.9, 2.5, 2.5,
2.7, 3.2, 3.3, 3.7, 3.8, 4.1, 4.5, 4.5, 4.5, 5.0,
5.1, 5.6, 5.7, 6.2, 6.3]) \
.reshape(-1, 1) #1
y = np.array([0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0,
0.0, 0.0, 1.0, 1.0, 0.0, 1.0, 1.0, 1.0, 1.0, 1.0,
1.0, 1.0, 1.0, 1.0, 1.0]) #2
model = LogisticRegression(penalty=None) #3
model.fit(x, y) #3
b1 = model.coef_.flatten()[0] #4
b0 = model.intercept_[0] #4
print(f"β₀={b0:.4f}, β₁={b1:.4f}") #5
This listing shows the complete scikit-learn fitting workflow, including reshaping the predictor, fitting the model, and reading β₀ and β₁. section 8.3
