IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals
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. What is the posterior mean of p?
The posterior is Beta(7, 23), because the Beta prior is conjugate to binomial data and adds successes and failures to the parameters. Its mean is 7/30. The sample proportion 5/20 would ignore the prior information, so it is incorrect.
- A7/30Correct
- B7/20
- C5/20
- D2/10
- 7/28
Explanation
Beta prior with binomial data gives posterior Beta(2+5, 8+15) = Beta(7, 23). Posterior mean = 7/(7+23) = 7/30. The option 5/20 is the sample proportion, ignoring the prior; 2/10 is the prior mean.
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
- Under squared error loss, the Bayes estimate of a parameter θ is which feature of the posterior distribution?
- 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…
- The proportion p of policies lapsing in a year has a Beta(2, 8) prior. In a sample of 20 policies, 6 lapse. What is the Bayesian estimate of…
- A prior for a Poisson mean λ is Gamma(α=4, β=2) (rate β). A portfolio has 10 claims observed over 3 years of exposure so that the sample dat…
- For an insurer in Pune, the claim count rate θ has a Gamma(4, 2) prior (rate parameterisation). Over 5 years, 12 claims are observed, Poisso…