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MathidiaMathematics Academy
Statistical inference
Lesson 36 of 75
Introduction

Likelihood

Objective. Construct likelihood and log-likelihood functions.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Likelihood measures how plausible different parameter values are given observed data, defined for independent observations x1,...,xn with density f(x;θ) as L(θ) = ∏ f(xᵢ;θ). The log-likelihood ℓ(θ) = Σ ln f(xᵢ;θ) is usually easier to work with since products become sums. For n Bernoulli(p) observations with k successes, L(p) = p^k(1−p)^(n−k), so the log-likelihood is ℓ(p) = k ln p + (n−k) ln(1−p), which is maximised to estimate p.

Worked example

Write the likelihood function for a single observation x from an Exponential(λ) distribution with density λe^(−λx).

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Illustrated card showing ?
  1. 1Start from the density f(x;λ) = λe^(−λx)
  2. 2With one observation, the likelihood equals the density itself
  3. 3Substitute the exponential density form
  4. 4State the likelihood function explicitly

Answer: L(λ) = λe^(−λx)

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Common mistake. A common mistake is treating the likelihood L(θ) as a probability distribution over θ and expecting it to integrate to one, but it is a function of θ for fixed data, not a density in θ. Likelihood compares relative plausibility, not probability of parameters.

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