91Ó°ÊÓ

A random variable is a factor with an indefinite quantity or a program that gives numbers to each study's results.

Suppose that μ and τ are random variables. Let the conditional distribution of given is the normal distribution with parameters \({\mu _{\scriptstyle0\atop\scriptstyle\,}}\) and \({\lambda _{\scriptstyle0\atop\scriptstyle\,}}\tau \) mean and precision, respectively, and let the marginal distribution of μ and τ is the gamma distribution with parameters and, then the joint distribution of and is called with the hyperparameters \({\mu _{\scriptstyle0\atop\scriptstyle\,}},{\lambda _{\scriptstyle0\atop\scriptstyle\,}},{\alpha _{\scriptstyle0\atop\scriptstyle\,}},{\beta _{\scriptstyle0\atop\scriptstyle\,}}{\rm{ }}\)

\({\mu _{\scriptstylecond\atop\scriptstyle\,}} = \frac{{{\mu _{\scriptstyle0\atop\scriptstyle\,}} + \sum\nolimits_{i = 1}^n {{y_{\scriptstylei\atop\scriptstyle\,}}{x_{\scriptstylei\atop\scriptstyle\,}}} }}{{{\lambda _{\scriptstyle0\atop\scriptstyle\,}} + \sum\nolimits_{i = 1}^n {{y_{\scriptstylei\atop\scriptstyle\,}}} }}\)

Because of how random variables \({y_{\scriptstylei\atop\scriptstyle\,}}\) are defined. The variance of the conditional distribution is

\({\sigma ^2}_{\scriptstylecond\atop\scriptstyle\,} = {\rm{ }}\frac{1}{{\tau \sum\nolimits_{i = 1}^n {{y_{\scriptstylei\atop\scriptstyle\,}}} }}.\)

\({\alpha _{\scriptstylecond\atop\scriptstyle\,}} = {\alpha _0} + \sum\limits_{i = 1}^n {\frac{{{y_{\scriptstylei\atop\scriptstyle\,}}}}{2} + \frac{1}{2}} \)

AND

\({\beta _{\scriptstylecond\atop\scriptstyle\,}} = {\beta _0} + \frac{{{\lambda _0}{{\left( {\mu - {\mu _0}} \right)}^2}}}{2} + \frac{1}{2}{\sum\limits_{i = 1}^n {{y_{\scriptstylei\atop\scriptstyle\,}}\left( {{X_{\scriptstylei\atop\scriptstyle\,}} - \mu } \right)} ^2}\)

Use the fact that \({y_{\scriptstylei\atop\scriptstyle\,}} = 1\)when they are from the normal distribution with mean\(\mu \) and precision, and\({y_{\scriptstylei\atop\scriptstyle\,}} = 0\) if \({X_{\scriptstylei\atop\scriptstyle\,}}\)theyare from the standard normal distribution.

The p.d.f.'s cancel the constant factor; thus, the probability is

\(Pr\left( {{y_{\scriptstylei\atop\scriptstyle\,}} = 1} \right){\rm{ }} = \frac{{\sqrt {{\rm{ }}\tau } \exp \left( { - {{\left( {{x_{\scriptstylei\atop\scriptstyle\,}} - \mu } \right)}^2}\tau /2} \right)}}{{\sqrt {{\rm{ }}\tau } \exp \left( { - {{\left( {{x_{\scriptstylei\atop\scriptstyle\,}} - \mu } \right)}^2}\tau /2} \right) + \exp \left( { - {x_{\scriptstylei\atop\scriptstyle\,}}^2/2} \right)}}{\rm{. }}\)

Gibbs Sampling is aMonte 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.

\(\left( {1.} \right)\,\)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\(\,i + 1\)

It would require a large simulation run to estimate the posterior distribution parameters. The distributions from (a)to(c) could be used to estimate the parameters. Start by splitting the data set into two subsets, preferably equal.

One could do that by assigning the probability of each observation. Next, start with \(\mu \,and{\rm{ }}\tau \)at their posterior means, given that the observations came from the corresponding distribution (with unknown parameters), using starting values of \({y_{\scriptstylei\atop\scriptstyle\,}}\). Obtain the sample parameters using distribution (a)to(c), and by averaging them, one obtains the estimate of the posterior distribution. Note that one should do that after burn-in and, as mentioned earlier, an extensive simulation run

From the exercise\({y_1} = 1\), when\({x_{\scriptstylei\atop\scriptstyle\,}}\) are from the normal distribution with mean \(\mu \,\)and precision \(\tau \) unknown. The fact that the posterior mean of is the posterior probability that\({y_{\scriptstylei\atop\scriptstyle\,}} = 1\)

because of the definition of the mean (\({y_{\scriptstylei\atop\scriptstyle\,}} = 0\) cancel the probability\(\Pr \left( {{y_{\scriptstylei\atop\scriptstyle\,}} = 0} \right)1\) ), the statement follows from the first sentence.

Hence,

((a) ) Normal distribution;

((b)) Gamma distribution;

((c)) \({y_{\scriptstylei\atop\scriptstyle\,}}\) can take values either \(0\,\,or\,\,1\);

((d)) The posterior mean is the posterior probability\({y_{\scriptstylei\atop\scriptstyle\,}} = 1\) .