IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals
A posterior distribution for a parameter θ is Normal with mean 40 and variance 25. What is the equal-tailed 95% Bayesian credible interval for θ? (Use z = 1.96.)
The interval is (30.2, 49.8). The posterior is Normal with standard deviation 5, so the 95% equal-tailed credible interval is the mean plus or minus 1.96 times 5, which is 40 plus or minus 9.8.
- A(30.2, 49.8)Correct
- B(35.0, 45.0)
- C(38.04, 41.96)
- D(15.5, 64.5)
- (32.7, 47.3)
Explanation
The standard deviation is sqrt(25)=5. The interval is 40 ± 1.96×5 = 40 ± 9.8, giving (30.2, 49.8). Option (38.04, 41.96) wrongly uses the variance-based half-width 1.96×1, i.e. treats the sd as 1.
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
- The posterior distribution of a claim rate parameter λ is Gamma with shape 12 and rate 4 (mean 3, variance 0.75). Under quadratic (squared e…
- Claim counts for a policyholder are Poisson with mean λ. The prior for λ is Gamma(α=2, rate β=1). In one year, 4 claims are observed. The po…
- 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:
- 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 Bayes…