Maximum Likelihood Estimation

Key idea

Choose parameters maximising the data’s probability.

In depth

Choose the parameter values that make the observed data most probable: maximise L(θ) = ΠP(xᵢ | θ), usually through the log-likelihood. It underlies many estimators, such as the sample mean for a normal distribution.

Worked example

For 7 heads in 10 flips, L(p) = p⁷(1 − p)³ is maximised at p̂ = 0.7.

Real-world uses

• Fitting a model to customer arrival data in telecoms and insurance

Common mistake

Treating the likelihood as a probability distribution over θ.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Probability & Statistics

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• Bayes’ Theorem – Update beliefs with evidence.
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