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Chapter 12: Q11E (page 838)

Consider, once again, the model described in Example \({\bf{7}}{\bf{.5}}{\bf{.10}}{\bf{.}}\) Assume that \({\bf{n = 10}}\) the observed values of \({{\bf{X}}_{\bf{1}}},...,{{\bf{X}}_{{\bf{1}}0}}\) are

\( - 0.92,\,\, - 0.33,\,\, - 0.09,\,\,\,0.27,\,\,\,0.50, - 0.60,\,1.66,\, - 1.86,\,\,\,3.29,\,\,\,2.30\).

a. Fit the model to the observed data using the Gibbs sampling algorithm developed in Exercise. Use the following prior hyperparameters: \({{\bf{\alpha }}_{\bf{0}}}{\bf{ = 1,}}{{\bf{\beta }}_{\bf{0}}}{\bf{ = 1,}}{{\bf{\mu }}_{\bf{0}}}{\bf{ = 0}}\,{\bf{and}}\,{\bf{ }}{{\bf{\lambda }}_{\bf{0}}}{\bf{ = 1}}\)

b. For each i, estimate the posterior probability that \({\rm{ }}{{\rm{x}}_i}\)came for the normal distribution with unknown mean and variance.

Short Answer

Expert verified

a) The Gibbs Sampling Algorithm: The steps of the algorithm are

\(\left( {1.} \right)\,\)Pick starting values \({x_2}^{\left( 0 \right)}\) for \(\,{x_2}\) , and let \(\,\,i = 0\,\)

b) Total of \(i = 1,2,...,10\)estimations of the posterior probability.

((a)) Use Gibbs Algorithm.

((b)) \(0.286,{\rm{ }}0.289,{\rm{ }}0.306,{\rm{ }}0.341,{\rm{ }}0.365,{\rm{ }}0.285,{\rm{ }}0.659,{\rm{ }}0.378,{\rm{ }}0.951,{\rm{ }}0.826\)

Step by step solution

01

(a) Definition of Gibbs sampling

Gibbs Sampling is a Monte Carlo Markov Chain method for estimating complex joint distributions that draw an instance from the distribution of each variable iteratively based on the current values of the other variables.

The Gibbs Sampling Algorithm: The steps of the algorithm are

\(\left( {1.} \right)\,\)Pick starting values \({x_2}^{\left( 0 \right)}\) for \(\,{x_2}\) , and let \(\,\,i = 0\,\)

\(\left( {2.} \right)\,\)let be a simulated value from the conditional distribution of \(\,{x_1}\)given that \(\,\,{X_1} = {x_2}^{\left( i \right)}\)

\(\left( {3.} \right)\,\)Let\(\,{x_2}^{\left( {i + 1} \right)\,\,}\,\) be a simulated value from the conditional distribution of \(\,{x_2}\) given that \(\,{X_1} = {x_1}^{\left( {i + 1} \right)}\)

\(\left( {4.} \right)\)Repeat steps \(\,2.\,\,and\,3.\) \(\,i\) where\(\,i + 1\)

Given data, the hyperparameters, and the Gibbs Algorithms, after running \(10\) Markov Chains with a total of \(N = 100000\) samples.

One may obtain the required values to estimate part(b) probabilities.

First, one may use for loop to fit the model described in the previous Exercise.

02

(b) Estimation probabilities

Total of estimations of the posterior probability.

Total of \(i = 1,2,...,10\)estimations of the posterior probability.

Using the method in the previous exercise, the 10 estimated probabilities are as follows

\(0.286,{\rm{ }}0.289,{\rm{ }}0.306,{\rm{ }}0.341,{\rm{ }}0.365,{\rm{ }}0.285,{\rm{ }}0.659,{\rm{ }}0.378,{\rm{ }}0.951,{\rm{ }}0.826\)

Hence,

((a)) Use Gibbs Algorithm.

((b)) \(0.286,{\rm{ }}0.289,{\rm{ }}0.306,{\rm{ }}0.341,{\rm{ }}0.365,{\rm{ }}0.285,{\rm{ }}0.659,{\rm{ }}0.378,{\rm{ }}0.951,{\rm{ }}0.826\)

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