IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals
For an insurer in Pune, the claim count rate θ has a Gamma(4, 2) prior (rate parameterisation). Over 5 years, 12 claims are observed, Poisson with mean θ per year. The loss is L(θ,a) = (θ − a)²/θ. The Bayes estimate under this loss is 1/E[1/θ | data]. What is it?
The estimate is 15/7, since the posterior is Gamma(16, 7) and 1/E[1/θ] = (α−1)/β = 15/7.
- A16/7Correct
- B17/7
- C2
- D16/9
- 8/3
Explanation
Posterior is Gamma(4+12, 2+5) = Gamma(16, 7). For Gamma(α, β), E[1/θ] = β/(α−1) = 7/15. Minimising E[(θ−a)²/θ] gives a = 1/E[1/θ] = 15/7. Check: expected loss derivative: −2E[1] + 2aE[1/θ]=0 gives a = 1/E[1/θ] = 15/7, so none of the stated values match unless computed carefully; the correct value 15/7 is not listed.
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