Problem Statement

Implement 5 iterations of the EM algorithm for a 1D two-component Gaussian Mixture Model (GMM) and return the final parameters.

E-step — compute responsibilities:

R[i,k] = π_k * N(x_i; μ_k, σ²_k) / Σ_j π_j * N(x_i; μ_j, σ²_j)

Use 1e-10 to avoid division by zero in the denominator.

M-step — update parameters:

N_k = Σ_i R[i,k]
μ_k = Σ_i R[i,k] * x_i / N_k
σ²_k = Σ_i R[i,k] * (x_i - μ_k)² / N_k
π_k = N_k / n

Add 1e-10 to N_k to prevent division by zero.

Function signature:

def gmm_em_5steps(X: np.ndarray, mu0: np.ndarray, var0: np.ndarray, pi0: np.ndarray) -> tuple:
    # returns (means, vars, weights) after 5 EM iterations

Input Format

Line 1: n — number of data points
Line 2: n space-separated floats — data values
Line 3: mu_0 mu_1 — initial means
Line 4: var_0 var_1 — initial variances
Line 5: pi_0 pi_1 — initial mixture weights

Output Format

Three lines (8 decimal places each):

1. Final means
2. Final variances
3. Final mixture weights

Example

Input:

6
1.0 1.5 2.0 8.0 8.5 9.0
1.5 8.5
1.0 1.0
0.5 0.5

Output:

1.50000000 8.50000000
0.16666667 0.16666667
0.50000000 0.50000000

Derivation

The EM algorithm maximizes the log-likelihood of a GMM by alternating between:

• E-step: compute the posterior probability (responsibility) that each data point belongs to each component.
• M-step: update means, variances, and weights as weighted averages using responsibilities as weights.

This guarantees non-decreasing log-likelihood convergence. With well-separated data (clusters at ~1.5 and ~8.5), the algorithm converges quickly.

Notes

• Use a small eps = 1e-10 for numerical stability in both E and M steps.
• 5 iterations is sufficient for well-separated clusters to converge.