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

In the Bayesian framework, the posterior density of a parameter θ given data x is proportional to which of the following?

The posterior density is proportional to the prior density multiplied by the likelihood. The marginal density of the data is just a normalising constant that does not depend on the parameter, so it can be ignored when identifying the posterior's shape.

  1. APrior density of θ multiplied by the likelihood of x given θCorrect
  2. BLikelihood of x given θ divided by the prior density of θ
  3. CPrior density of θ plus the likelihood of x given θ
  4. DLikelihood of x given θ alone, since the prior is a constant
  5. Prior density of θ divided by the likelihood of x given θ

Explanation

Bayes' theorem gives f(θ|x) = f(x|θ)f(θ)/f(x). The denominator f(x) does not depend on θ, so the posterior is proportional to prior times likelihood. The other options combine the terms in ways that do not follow from Bayes' theorem.

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