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What is Maximum Likelihood Estimation?

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Answer

  • Choose parameter values that make the observed data most probable — i.e., maximize the likelihood function L(θ)  =  P(data  ∣  θ)L(\theta)\; = \;P(\mathrm{data}\; \mid \;\theta).
  • In practice, maximize log L (numerically stable, turns products into sums).
  • MLEs are consistent, asymptotically normal, and efficient under regularity conditions.
  • Add regularization/priors to obtain MAP estimation (Bayesian penalty).
Check yourself — multiple choice
  • MLE minimizes the likelihood
  • MLE maximizes the probability of observed data given parameters
  • MLE requires a prior
  • MLE always uses closed-form solutions

MLE  =  argmax⁡θ\mathrm{MLE}\; = \;\operatorname{argmax}_{\theta} P(data  ∣  θ)P(\mathrm{data}\; \mid \;\theta).

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