Problem Statement

Given n observations of a d-dimensional multivariate Gaussian, compute the Maximum Likelihood Estimates (MLE) for the mean vector and covariance matrix.

Formulas (biased MLE, divide by n):

μ_MLE = (1/n) Σ x_i

Σ_MLE = (1/n) Σ (x_i - μ)(x_i - μ)ᵀ = (X_c)ᵀ X_c / n

where X_c = X - μ is the mean-centered data matrix.

Function signature:

def mle_mvg(X: np.ndarray) -> tuple:
    # returns (mean, cov)
    # mean: shape (d,), cov: shape (d, d)

Input Format

Line 1: n d — number of observations and dimensions
Lines 2..n+1: d space-separated floats per row

Output Format

Line 1: d space-separated mean values (8 decimal places)
Lines 2..d+1: d × d covariance matrix rows (8 decimal places each)

Example

Input:

4 2
1.0 2.0
3.0 4.0
5.0 6.0
7.0 8.0

Output:

4.00000000 5.00000000
5.00000000 5.00000000
5.00000000 5.00000000

Derivation

The MLE for the multivariate Gaussian is obtained by maximizing the log-likelihood:

log L(μ, Σ) = -n/2 log|Σ| - 1/2 Σ_i (x_i - μ)ᵀ Σ⁻¹ (x_i - μ) + const

Setting the derivative w.r.t. μ to zero gives μ_MLE = sample mean. Substituting back and differentiating w.r.t. Σ gives the biased covariance (divide by n).

Notes

• The MLE covariance is biased (divides by n). The unbiased sample covariance divides by n-1.
• Use X_c.T @ X_c / n rather than numpy.cov (which uses ddof=1 by default).
• The result is always symmetric positive semi-definite.