91Ó°ÊÓ

Recall what is stated in the mentioned example and the normal and gamma distribution probability density functions. Then, the required product is a product of

\(\exp \left\{ { - \frac{{{u_0}{{\left( {\psi - {\psi _0}} \right)}^2}}}{2} - \sum\limits_{i = 1}^p {{T_i}} \left( {{\beta _0} + \frac{{{n_i}{{\left( {{\mu _i} - \mathop {{y_i}}\limits^\_ } \right)}^2} + {w_i} + \lambda {{\left( {{\mu _i} - \psi } \right)}^2}}}{2}} \right)} \right\}\)

and the following expression

\({\lambda ^{p/2 + {\gamma _0} - 1}}\exp \left( { - \lambda {\delta _0}} \right)\,\,\mathop \Pi \limits_{i = 1}^p T_i^{{\alpha _0} + \left( {{n_i} + 1} \right)/2 - 1},\)

and \({w_i}\) is given in the same way as in the example

\({w_i} = \sum\limits_{j = 1}^{{n_i}} {{{\left( {{y_{ij}} - \mathop {{y_i}}\limits^\_ } \right)}^2}} ,\,\,\,\,i = 1,2,...,p.\)

The second term corresponds to the prior distribution.

As mentioned in the hint, the conditional distributions stay practically the same as in the example.

If they look at a product as a function of \({T_i}\), then you get a gamma distribution with parameters

\({\alpha _0} + \frac{{{n_i} + 1}}{2}\,\) and \(\,\,{\beta _0} + \frac{{{n_i}{{\left( {{\mu _i} - \mathop {{y_i}}\limits^\_ } \right)}^2} + {w_i} + \lambda {{\left( {{\mu _i} - \psi } \right)}^2}}}{2}\)

Next, as a function of \(\psi \)by writing it in a form widely known, one gets a normal distribution with parameters

\(\frac{{{u_0}{\psi _0} + \lambda \sum\nolimits_{i = 1}^p {\,\,} {\mu _i}}}{{{u_0} + \lambda \sum\nolimits_{i = 1}^p {{T_i}\,} }}\,\) and \({u_0} + \lambda \sum\limits_{i = 1}^p {{T_i}} \,.\)

Before commenting on the distribution of given all others, notice that the product above looks like a probability density function of a normal distribution when it is a function of for the same reason. The mean and precision are given with

\(\frac{{{n_i}\mathop {{y_i}}\limits^\_ + \lambda \psi }}{{{n_i} + {\lambda _0}}}\) and \(\,{u_0}{T_i}\,\left( {{n_i} - \lambda } \right).\)

Finally, if the product is seen as a function of \(\lambda \), it approximately probability density function of gamma distribution with parameters

\(\frac{p}{2} + {\gamma _0}\,\) and \({\delta _0} + \frac{1}{2}\sum\limits_{i = 1}^p {{T_i}} {\left( {{\mu _i} - \psi } \right)^2}.\)

The parameters that should be used are given in the exercise, and the others are

\({n_1} = 20\) for beef,

\({n_2} = 17\) for meat,

\({n_3} = 17\) for poultry,

\({n_4} = 9\) for specialty.

Means \({\mu _i}\) \(i = 1,2,3,4\)correspond to the indices of \({n_i}\), \(i = 1,2,3,4\)

The code used for this simulation uses N = 20000 Markov chains with I = 100000 iterations/steps. It means that there are a total of I = 100000 parameter vector from which the result is obtained. Note that the given code should be changed as the initial parameters are different. The estimated posterior means of \({\mu _i}\), \(i = 1,2,3,4\) are, respectively, 157, 158.6, 118.9, and 160.4. Similarly, the estimated posterior means for the \(1/{T_i}\) \(i = 1,2,3,4\) are respectively, 487 598.9, 479.44, and 548.1

Here, the result of part(a), part(b) and part(c)

1. The product of the two functions uses parameters given in the exercise and distribution \(\lambda \).
2. The conditional distribution of \(\lambda \) all other parameters is gamma distribution.
3. The estimated posterior means of \({\mu _i}\) \(i = 1,2,3,4\) are, respectively, 157 158.6, 118.9, and 160.4.
4. The estimated posterior means for the \(1/{T_i}\) \(i = 1,2,3,4\) are, respectively, 487 598.9, 479.44, and 548.1.