Q6

First we generate the data using the information given in the question. The 10 theta values I have chosen as 1, 2, …, 10 and the population variance as 1. I generate 25 sample points from each population and display first few readings and a box-plot of the same.

Type 1 Type 2 Type 3 Type 4 Type 5 Type 6 Type 7 Type 8 Type 9 Type 10
0.4395244 0.3133067 3.253318 5.025571 4.289593 5.348050 7.787739 6.904004 11.198810 10.059750
0.7698225 2.8377870 2.971453 3.715227 5.256884 6.235387 7.769042 8.037788 10.312413 9.295404
2.5587083 2.1533731 2.957130 2.779282 4.753308 6.077961 7.332203 8.310481 8.734855 9.282782
1.0705084 0.8618631 4.368602 4.181304 4.652457 5.038143 5.991623 8.436523 9.543194 10.884650
1.1292877 3.2538149 2.774229 3.861109 4.048381 5.928692 6.880547 7.541635 8.585660 8.984407
2.7150650 2.4264642 4.516471 4.005764 4.954972 7.444551 6.719605 6.936674 8.523753 11.955294

Gibbs Sampling using R program

We run a Markov Chain of size 10,000 and choose the last 5000 values for estimation of each population mean.

1st choice of Hyperparameter Values

Here we choose the hyperparamters as moderately vague and the mean for the Normal distribution is chosen to be zero. We display estimates of all 10 means whose true values are 1, 2, 3, …, 10.

# Hyper parameters

u0 <- 0    # Mean of the Normal distribution  
v0 <- 10  # Variance of the Normal distribution
a1 <- 0.1  # as defined in book
b1 <- 0.1  # as defined in book
a2 <- 0.1  # as defined in book
b2 <- 0.1  # as defined in book
The estimates are:
0.9830428 
2.114153 
3.017667 
4.284433 
4.722796 
5.766093 
7.205333 
7.850506 
9.08568 
9.854426

We obtain reasonable estimates.

2nd choice of Hyperparameter Values

Next we use the same vague priors but use a larger hyperparameter for the mean of the Normal distribution to see if there is some difference.

# Hyper parameters

u0 <- 50    # Mean of the Normal distribution  
v0 <- 10  # Variance of the Normal distribution
a1 <- 0.1  # as defined in book
b1 <- 0.1  # as defined in book
a2 <- 0.1  # as defined in book
b2 <- 0.1  # as defined in book
The estimates are:
0.9662683 
2.102985 
3.009338 
4.282624 
4.722492 
5.772413 
7.216608 
7.8611 
9.105483 
9.87523

We obtain similar estimates as before.

3rd choice of Hyperparameter Values

Next we use the same vague priors but use a much larger hyperparameter for the mean of the Normal distribution to see if there is some difference.

# Hyper parameters

u0 <- 1000    # Mean of the Normal distribution  
v0 <- 10  # Variance of the Normal distribution
a1 <- 0.1  # as defined in book
b1 <- 0.1  # as defined in book
a2 <- 0.1  # as defined in book
b2 <- 0.1  # as defined in book
The estimates are:
0.9651634 
2.101969 
3.008228 
4.281849 
4.721804 
5.770914 
7.215595 
7.860368 
9.104549 
9.874419

We obtain similar estimates as before. This indicates that the u0 hyperparameter does not effec the results. Next we increase the vagueness of priors.

4th choice of Hyperparameter Values

We use some very vague priors.

# Hyper parameters

u0 <- 0    # Mean of the Normal distribution  
v0 <- 1000000  # Variance of the Normal distribution
a1 <- 0.000001  # as defined in book
b1 <- 0.000001  # as defined in book
a2 <- 0.000001  # as defined in book
b2 <- 0.000001  # as defined in book
The estimates are:
0.9651365 
2.100891 
3.008035 
4.282231 
4.721153 
5.771179 
7.213425 
7.860911 
9.105514 
9.875362

There is no noticeable difference from previous estimates. Next we choose priors that are not vague.

5th choice of Hyperparameter Values
# Hyper parameters

u0 <- 0    # Mean of the Normal distribution  
v0 <- 0.01  # Variance of the Normal distribution
a1 <- 100  # as defined in book
b1 <- 100  # as defined in book
a2 <- 100  # as defined in book
b2 <- 100  # as defined in book
The estimates are:
0.9466822 
2.07878 
2.970404 
4.218997 
4.657986 
5.681872 
7.109531 
7.753166 
8.97402 
9.730353

We do not see appreciable change in results.

Gibbs Sampling using JAGS

We run the sampler and produce the estimates.

Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 250
   Unobserved stochastic nodes: 13
   Total graph size: 520

Initializing model

Now we check the estimates.

The estimates are:
1.000002 
1.999997 
2.999999 
3.999991 
4.999999 
5.999991 
6.999997 
7.999997 
8.999985 
10

The results are way more accurate than those obtained by the previous Gibbs Sampler.

END