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IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals

A prior distribution for a parameter θ is combined with data x. Which statement correctly describes the Bayesian relationship between the posterior density, the prior density and the likelihood?

The posterior density is proportional to the prior density multiplied by the likelihood, because the denominator f(x) is just a constant in θ. Normalising the product so it integrates to one gives the full posterior distribution.

  1. APosterior is proportional to prior multiplied by likelihoodCorrect
  2. BPosterior is proportional to prior divided by likelihood
  3. CPosterior is proportional to likelihood only, with the prior ignored
  4. DPosterior is proportional to prior plus likelihood
  5. Posterior equals the likelihood normalised over the data x

Explanation

Bayes' theorem gives f(θ|x) = f(x|θ)f(θ)/f(x), where f(x) does not depend on θ. So the posterior is proportional to likelihood times prior. The other options either drop the prior, combine the terms incorrectly or normalise over the wrong variable.

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