IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals
The proportion p of policies lapsing in a year has a Beta(2, 8) prior. In a sample of 20 policies, 6 lapse. What is the Bayesian estimate of p under quadratic loss?
The posterior is Beta(8, 22), obtained by adding 6 lapses to the first parameter and 14 non-lapses to the second. The quadratic loss estimate is the posterior mean 8/30, which is 0.267.
- A0.267Correct
- B0.300
- C0.200
- D0.280
- 0.233
Explanation
The posterior is Beta(2 + 6, 8 + 14) = Beta(8, 22). The posterior mean is 8/30 = 0.267. The 0.300 is the sample proportion 6/20 and 0.200 is the prior mean 2/10. Option 0.280 is a mistaken mode-type figure and is not the mean.
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