Gaussian Mixture Model

Task

Implement class GaussianMixtureModel(initial_weights, initial_means, initial_variances, max_iterations, tolerance, epsilon). Implement a diagonal-covariance Gaussian mixture model trained with expectation-maximization, including stable responsibilities, parameter updates, convergence tracking, probabilities, predictions, and total log-likelihood.

Requirements

• Copy initial_weights, initial_means, and initial_variances so caller-owned arrays are not mutated.
• Clamp every variance to at least epsilon during initialization and after each M-step.
• e_step(x) computes responsibilities in log space and returns one normalized row per example.
• m_step(x, responsibilities) updates weights, means, and diagonal variances from component effective counts.
• score(x) returns the total data log-likelihood using a stable log-sum-exp calculation.
• fit(x) performs at most max_iterations E-steps and M-steps and appends the post-update total log-likelihood to log_likelihood_history.
• After at least two updates, converge when the absolute change in consecutive log-likelihood values is less than or equal to tolerance.
• Store n_iterations, converged, and final responsibilities after fitting.
• predict_proba(x) returns responsibilities and predict(x) returns the highest-responsibility component; ties choose the lowest component index.

Example

model = GaussianMixtureModel(
    np.array([0.5, 0.5]),
    np.array([[0.0], [10.0]]),
    np.array([[2.0], [2.0]]),
    max_iterations=20,
    tolerance=1e-8,
    epsilon=1e-6,
)
model.fit(np.array([[0.0], [2.0], [8.0], [10.0]]))
model.predict(np.array([[1.0], [9.0], [5.0]]))
# array([0, 1, 0])