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

Claim counts for a policyholder are Poisson with mean λ. The prior for λ is Gamma(α=2, rate β=1). In one year, 4 claims are observed. The posterior is Gamma with parameters and the posterior mean is:

The posterior is Gamma(6, 2) with mean 3. The shape increases by the observed claims, 2 plus 4, and the rate increases by the one year of exposure, 1 plus 1, so the mean is 6 divided by 2.

  1. AGamma(6, 2); mean 3Correct
  2. BGamma(4, 1); mean 4
  3. CGamma(6, 1); mean 6
  4. DGamma(2, 5); mean 0.4
  5. Gamma(6, 5); mean 1.2

Explanation

Posterior shape = α + x = 2+4 = 6 and rate = β + n = 1+1 = 2. The mean is 6/2 = 3. Gamma(6,1) wrongly forgets to add the exposure to the rate.

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