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

Which statement about an improper prior such as f(θ) ∝ 1 for a normal mean θ (variance known) is correct?

An improper flat prior does not integrate to a finite value, but combined with normal data it yields a proper posterior N(x̄, σ²/n). So it is not always improper, and the posterior mean equals the sample mean.

  1. AIt cannot integrate to a finite value, yet it can still give a proper posteriorCorrect
  2. BIt always gives an improper posterior
  3. CIt is conjugate only when the variance is unknown
  4. DIt makes the posterior mean differ from the sample mean
  5. It is a Beta prior with parameters 0 and 0

Explanation

A flat prior over the real line has infinite integral, so it is improper. Multiplying by the normal likelihood gives a posterior proportional to the likelihood, which is a proper N(x̄, σ²/n). So the posterior mean equals the sample mean.

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