IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals
A prior for a proportion θ is Beta(2, 8). After observing 6 successes in 20 trials, what is the posterior mean of θ, the centre of the Bayesian estimate under quadratic loss?
The posterior mean is 8/30, about 0.267. The posterior is Beta(8, 22) because 6 successes are added to 2 and 14 failures to 8. Under quadratic loss the Bayes estimate is this posterior mean.
- A8/30Correct
- B8/28
- C6/20
- D2/10
- 6/30
Explanation
The posterior is Beta(2+6, 8+14) = Beta(8, 22). Its mean is 8/(8+22) = 8/30. The sample proportion 6/20 ignores the prior, and 8/28 omits the failures' contribution wrongly by using 8+20.
Did you get it right without looking?
One question tells you little. A timed set on Bayesian inference: priors, posteriors, loss functions and credible intervals shows your real accuracy, how long you take and where you lose marks.
More Bayesian inference: priors, posteriors, loss functions and credible intervals questions
- Observations x1,...,x_n are normal with unknown mean μ and known variance 16. The prior for μ is N(50, 16). A sample of n = 4 has mean 58. W…
- 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,…
- Under quadratic (squared error) loss, the Bayesian estimate of a parameter θ is:
- Which statement about a 95% Bayesian credible interval (a, b) for θ is correct?
- A posterior distribution for a claim rate λ is Exponential with mean 4 (rate 0.25). Under absolute error loss L(λ,a) = |λ − a|, the Bayes es…
- A claim-frequency parameter p (probability a policy has a claim) is given a Beta(2, 8) prior. In a sample of 20 policies, 5 have a claim. Wh…