IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals
The posterior of θ is discrete: P(θ=1)=0.2, P(θ=2)=0.5, P(θ=4)=0.3. Under all-or-nothing (0-1) loss, the Bayes estimate of θ is:
Under all-or-nothing loss the Bayes estimate is the posterior mode. The most probable value is 2, with probability 0.5. The posterior mean of 2.4 would apply only under quadratic loss.
- A2.0Correct
- B2.4
- C1.0
- D4.0
- 3.0
Explanation
All-or-nothing loss gives the posterior mode, which is the value with the highest probability, namely θ=2 (0.5). The mean is 0.2+1.0+1.2 = 2.4, which is the answer under quadratic loss. The median is also 2 but the mode is the rule being tested.
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