IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals
The posterior distribution of a parameter θ is Gamma with shape 6 and rate 2 (mean 3, variance 1.5). Under quadratic loss L(θ,a) = (θ − a)², what is the Bayes estimate of θ?
The Bayes estimate is 3. Under quadratic (squared error) loss the optimal point estimate is the posterior mean, and a Gamma distribution with shape 6 and rate 2 has mean 6/2 = 3. The variance of 1.5 is not the estimate.
- A2
- B2.5
- C3Correct
- D1.5
- 6
Explanation
Under quadratic loss the Bayes estimate is the posterior mean. The mean of Gamma(6, 2) is 6/2 = 3. The value 1.5 is the posterior variance, which is not an estimate under this loss.
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