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MLE for Normal(μ,  σ2)\mathrm{Normal}({\mu}, \;{\sigma}^{2}) — result.

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Answer

  • μ̂ = X̄ (unbiased, minimum variance). σ̂² = (1/n)  ⋅  Σ(Xi  −  X)2(1 / n)\; \cdot \;{\Sigma}(X_{i}\; - \;X)^{2} — the biased MLE.
  • Bias-corrected version divides by n-1 (sample variance).
  • For large n, both converge to σ2{\sigma}^{2}.
  • Deep-learning tie-in: minimizing MSE under Gaussian noise = MLE with fixed σ.
  • Softmax cross-entropy = MLE for a categorical distribution.
  • Almost every parametric ML training objective is an MLE.
Check yourself — multiple choice
  • Only Bayesian
  • μ̂ = X̄; σ̂²_MLE = (1/n) Σ(X  −  X)2{\Sigma}(X\; - \;X)^{2} (biased); sample variance divides by n-1 to correct — MSE = MLE under Gaussian noise
  • Random
  • μ̂ = median

MLE Normal: μ̂=X̄, σ̂²_MLE = (1/n)Σ(X−X)2{\Sigma}(X - X)^{2} (biased); MSE ≡ Gaussian MLE.

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