Bayesian mortality estimation for a small group and a planned hierarchical model
A small closed group life portfolio has 1,200 life-years of exposure and 8 deaths. The chief actuary proposes a Bayesian analysis of the force of mortality mu, using an informative Gamma(2, 200) prior based on a larger industry table; this prior has mean 0.01 and standard deviation about 0.0071. The likelihood is Poisson with mean 1,200 mu. A colleague prefers the maximum likelihood estimate, 8/1,200 = 0.00667, because it is unbiased and objective. You need to produce a posterior mean and 95% credible interval, assess prior sensitivity, and plan an MCMC analysis for a later hierarchical model that will borrow strength across 20 similar groups. The hierarchical model will have group-specific log-mu parameters with a common normal prior, so conjugacy will be lost.
1. Derive the posterior distribution, posterior mean, and an exact 95% credible interval. Compare with the MLE.
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The prior is Gamma(shape 2, rate 200), density proportional to mu^(2-1) exp(-200 mu). The Poisson likelihood for 8 deaths with exposure 1,200 is proportional to mu^8 exp(-1,200 mu). Multiplying gives a Gamma(2 + 8, 200 + 1,200) = Gamma(10, 1400) posterior. The posterior mean is 10/1400 = 0.007143.
For the exact interval, 2 times 1400 times mu has a chi-square distribution with 20 degrees of freedom. The lower limit is qchisq(0.025, 20)/(2 times 1400) = 9.591/2800 = 0.00343. The upper limit is qchisq(0.975, 20)/2800 = 34.170/2800 = 0.01220. The 95% credible interval is therefore about 0.0034 to 0.0122. The MLE is 0.00667. The prior mean of 0.01 is higher than the MLE, so the posterior mean is pulled above the MLE. The interval is wide because 8 deaths carry limited information; the data have not overwhelmed the prior.
2. Assess sensitivity to the prior. What would change if the prior were Gamma(1, 100) or Gamma(20, 2000)?
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Gamma(1, 100) has the same prior mean 0.01 but is much weaker. The posterior becomes Gamma(1 + 8, 100 + 1200) = Gamma(9, 1300), with mean 9/1300 = 0.006923. The exact 95% interval uses a chi-square with 18 degrees of freedom: lower = qchisq(0.025, 18)/2600 = 8.231/2600 = 0.00317; upper = qchisq(0.975, 18)/2600 = 31.526/2600 = 0.01213. This is close to the MLE, showing that the weaker prior matters less.
Gamma(20, 2000) also has mean 0.01 but is much stronger. The posterior is Gamma(28, 3200), with mean 28/3200 = 0.00875 and a narrower interval. The prior now dominates the 8 deaths and pulls the estimate noticeably upward. The lesson is that prior strength, not just prior mean, drives sensitivity. Check the prior predictive distribution for the number of deaths: for Gamma(2, 200), the predictive is negative binomial with mean 12, and 8 deaths is plausible but on the low side. Report results under at least two priors and consider a weakly informative prior or the hierarchical model.
3. Outline an MCMC scheme for the hierarchical model, including diagnostics and how you would check convergence. Why is conjugacy lost?
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Write the hierarchical model as D_j ~ Poisson(E_j mu_j), log mu_j = alpha + u_j, and u_j ~ Normal(0, sigma^2), with priors on alpha and sigma. Conjugacy is lost because the normal prior on log mu_j is not conjugate to the Poisson likelihood for mu_j, and the variance components alpha and sigma do not have closed-form conditional posteriors. A Gibbs sampler alone is therefore not available.
Use Metropolis-Hastings within Gibbs: update each u_j with a random-walk proposal centred on its current value, update alpha with a normal random walk, and update sigma with a positive random walk or slice sampler. Alternatively, use Hamiltonian Monte Carlo in Stan. Run at least four chains from overdispersed starting values, with 2,000 iterations after warm-up. Check trace plots, Rhat below 1.01, effective sample size above 400, autocorrelation and divergent transitions. Use posterior predictive checks for the group counts and compare the hierarchical estimates with no-pooling and complete-pooling estimates to confirm that shrinkage is sensible. Report sensitivity to the prior on sigma.