IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals
The prior for θ is Beta(2, 6). In 10 independent trials with success probability θ, 4 successes are observed. Under quadratic (squared error) loss, what is the Bayes estimate of θ?
The Bayes estimate is 0.333. The posterior is Beta(6, 12) after adding 4 successes and 6 failures to the prior parameters, and under squared error loss the estimate is the posterior mean, 6/18.
- A0.250
- B0.333Correct
- C0.400
- D0.300
- 0.350
Explanation
Posterior is Beta(2+4, 6+6) = Beta(6, 12). Under quadratic loss the Bayes estimate is the posterior mean = 6/18 = 0.333. The option 0.400 is the sample proportion, ignoring the prior; 0.250 is the prior mean.
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