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Problem 2.6 MLE for a multivariate Gaussian

In this problem you will derive the ML estimate for a multivariate Gaussian. Given samples {x1,⋯ ,xN}\{x_1, \cdots, x_N\},

(a) Derive the ML estimate of the mean μ\mu of a multivariate Gaussian.

You may find the following vector and matrix derivatives helpful:

  • ∂∂xaTx=a\frac{\partial}{\partial x} a^T x = a, for vectors x,a∈Rdx, a \in \mathbb{R}^d.
  • ∂∂xxTAx=Ax+ATx\frac{\partial}{\partial x} x^T A x = Ax + A^T x, for vector x∈Rdx \in \mathbb{R}^d and matrix A∈Rd×dA \in \mathbb{R}^{d \times d}.
  • ∂∂Xlog⁡∣X∣=X−T\frac{\partial}{\partial X} \log |X| = X^{-T}, for a square matrix XX.
  • ∂∂Xtr(AX−1)=∂∂Xtr(X−1A)=−(X−TATX−T)\frac{\partial}{\partial X} \text{tr}(AX^{-1}) = \frac{\partial}{\partial X} \text{tr}(X^{-1}A) = -(X^{-T} A^T X^{-T}), for matrices A,XA, X.

Hint: remember Σ\Sigma is symmetric!