Fig. 1. Marginal posterior (π_post) and partial posterior (π_pp) distributions for μ and σ², along with values of the cMLE and MLE estimators for the data of the normal example.

Fig. 2. Marginal conditional posterior and partial posterior distributions for μ and σ² as well as CP and PPP distributions for t using the different algorithms. The first two rows correspond to the conditional distributions, the first row is derived using Gibbs and the second using M–H. The third row shows all three algorithms for the partial posterior distributions.

Fig. 3. Outputs from Gibbs. The pictures graph the mean and the realizations from the posteriors against iteration number for several values of δ. The traces are only shown after the 5000 burning iterations.

Fig. 4. Outputs from Gibbs. The pictures graph the mean and the realizations from the posteriors against iteration number for several values of δ. The traces are only shown after the 5000 burning iterations.

Fig. 5. Outputs from Metropolis–Hastings. The pictures graph the mean and the realizations from the posteriors against iteration number for several values of δ. The traces are only shown after the 5000 burning iterations.

Fig. 6. Outputs from Metropolis–Hastings. The pictures graph the mean and the realizations from the posteriors against iteration number for several values of δ. The traces are only shown after the 5000 burning iterations.

Fig. 7. Partial posterior distributions for μ. The pictures graph the mean and the generated values of μ against iteration number for each of the three algorithms.

Fig. 8. Partial posterior distributions for σ². The pictures graph the mean and the generated values of σ² against iteration number for each of the three algorithms.

Fig. 9. Approximate conditional distributions for ∑logx_i given ∑x_i when α=1 for samples generated from a Ga(3,3) (left) and a Ga(1,1) (right) distributions. Vertical lines are the observed values of ∑logx_i

Some characteristics of the conditional and partial posterior predictive distributions for μ and σ² (CP and PPP) for T=max{X₁,…,X_n} using the different algorithms

In parenthesis we show the acceptance rate for M–H.

Mean of conditional distribution (11) and associated p-value for the Ga(3,3) sample (with MLE=0.936) and various values of β and δ